A Short Account of the History of Mathematics · W. W. Rouse Ball
He was the illegitimate son of a lawyer of Milan, and was educated at the universities of Pavia and Padua. After taking his degree he commenced life as a doctor, and practised his profession at Sacco and Milan from 1524 to 1550; it was during this period that he studied mathematics and published his chief works. After spending a year or so in France, Scotland, and England, he returned to Milan as professor of science, and shortly afterwards was elected to a chair at Pavia. Here he divided his time between debauchery, astrology, and mechanics. His two sons were as wicked and passionate as himself: the elder was in 1560 executed for poisoning his wife, and about the same time Cardan in a fit of rage cut off the ears of the younger who had committed some offence; for this scandalous outrage he suffered no punishment, as the Pope Gregory XIII. granted him protection. In 1562 Cardan moved to Bologna, but the scandals connected with his name were so great that the university took steps to prevent his lecturing, and only gave way under pressure from Rome. In 1570 he was imprisoned for heresy on account of his having published the horoscope of Christ, and when released he found himself so generally detested that he determined to resign his chair. At any rate he left Bologna in 1571, and shortly afterwards moved to Rome. Cardan was the most distinguished astrologer of his time, and when he settled at Rome he received a pension in order to secure his services as astrologer to the papal court. This proved fatal to him for, having foretold that he should die on a particular day, he felt obliged to commit suicide in order to keep up his reputation--so at least the story runs.
The chief mathematical work of Cardan is the Ars Magna published at Nuremberg in 1545. Cardan was much interested in the contest between Tartaglia and Fiore, and as he had already begun writing this book he asked Tartaglia to communicate his method of solving a cubic equation. Tartaglia refused, whereupon Cardan abused him in the most violent terms, but shortly afterwards wrote saying that a certain Italian nobleman had heard of Tartaglia's fame and was most anxious to meet him, and begged him to come to Milan at once. Tartaglia came, and though he found no nobleman awaiting him at the end of his journey, he yielded to Cardan's importunity, and gave him the rule, Cardan on his side taking a solemn oath that he would never reveal it. Cardan asserts that he was given merely the result, and that he obtained the proof himself, but this is doubtful. He seems to have at once taught the method, and one of his pupils Ferrari reduced the equation of the fourth degree to a cubic and so solved it.
When the Ars Magna was published in 1545 the breach of faith was made manifest. Tartaglia not unnaturally was very angry, and after an acrimonious controversy he sent a challenge to Cardan to take part in a mathematical duel. The preliminaries were settled, and the place of meeting was to be a certain church in Milan, but when the day arrived Cardan failed to appear, and sent Ferrari in his stead. Both sides claimed the victory, though I gather that Tartaglia was the more successful; at any rate his opponents broke up the meeting, and he deemed himself fortunate in escaping with his life. Not only did Cardan succeed in his fraud, but modern writers have often attributed the solution to him, so that Tartaglia has not even that posthumous reputation which at least is his due.
The Ars Magna is a great advance on any algebra previously published. Hitherto algebraists had confined their attention to those roots of equations which were positive. Cardan discussed negative and even complex roots, and proved that the latter would always occur in pairs, though he declined to commit himself to any explanation as to the meaning of these "sophistic" quantities which he said were ingenious though useless. Most of his analysis of cubic equations seems to have been original; he shewed that if the three roots were real, Tartaglia's solution gave them in a form which involved imaginary quantities. Except for the somewhat similar researches of Bombelli a few years later, the theory of imaginary quantities received little further attention from mathematicians until John Bernoulli and Euler took up the matter after the lapse of nearly two centuries. Gauss first put the subject on a systematic and scientific basis, introduced the notation of complex variables, and used the symbol i , which had been introduced by Euler in 1777, to denote the square root of (-1): the modern theory is chiefly based on his researches.
Cardan established the relations connecting the roots with the coefficients of an equation. He was also aware of the principle that underlies Descartes's "rule of signs," but as he followed the custom, then general, of writing his equations as the equality of two expressions in each of which all the terms were positive he was unable to express the rule concisely. He gave a method of approximating to the root of a numerical equation, founded on the fact that, if a function have opposite signs when two numbers are substituted in it, the equation obtained by equating the function to zero will have a root between these two numbers.
Cardan's solution of a quadratic equation is geometrical and substantially the same as that given by Alkarismi. His solution of a cubic equation is also geometrical, and may be illustrated by the following case which he gives in chapter xi. To solve the equation x^3+6x=20 (or any equation of the form x^3+qx=r ), take two cubes such that the rectangle under their respective edges is 2 (or 1/3q ) and the difference of their volumes is 20 (or r ). Then x will be equal to the difference between the edges of the cubes. To verify this he first gives a geometrical lemma to shew that, if from a line AC a portion CB be cut off, then the cube on AB will be less than the difference between the cubes on AC and BC by three times the right parallelepiped whose edges are respectively equal to AC , BC , and AB --this statement is equivalent to the algebraical identity (a-b)^3=a^3-b^3-3ab(a-b) --and the fact that x satisfies the equation is then obvious. To obtain the lengths of the edges of the two cubes he has only to solve a quadratic equation for which the geometrical solution previously given sufficed.
Like all previous mathematicians he gives separate proofs of his rule for the different forms of equations which can fall under it. Thus he proves the rule independently for equations of the form x^3+px=q , x^3=px+q , x^3+px+q=0 , and x^3+q=px . It will be noticed that with geometrical proofs this was the natural course, but it does not appear that he was aware that the resulting formulae were general. The equations he considers are numerical.
Shortly after Cardan came a number of mathematicians who did good work in developing the subject, but who are hardly of sufficient importance to require detailed mention here. Of these the most celebrated are perhaps Ferrari and Rheticus.
Ferrari. Ludovico Ferraro, usually known as Ferrari, whose name I have already mentioned in connection with the solution of a biquadratic equation, was born at Bologna on Feb. 2, 1522, and died on Oct. 5, 1565. His parents were poor and he was taken into Cardan's service as an errand boy, but was allowed to attend his master's lectures, and subsequently became his most celebrated pupil. He is described as "a neat rosy little fellow, with a bland voice, a cheerful face, and an agreeable short nose, fond of pleasure, of great natural powers," but "with the temper of a fiend." His manners and numerous accomplishments procured him a place in the service of the Cardinal Ferrando Gonzago, where he managed to make a fortune. His dissipations told on his health, and he retired in 1565 to Bologna where he began to lecture on mathematics. He was poisoned the same year either by his sister, who seems to have been the only person for whom he had any affection, or by her paramour.
Such work as Ferrari produced is incorporated in Cardan's Ars Magna or Bombelli's Algebra, but nothing can be definitely assigned to him except the solution of a biquadratic equation. Colla had proposed the solution of the equation x^4+6x^2+36=60x as a challenge to mathematicians: this particular equation had probably been found in some Arabic work. Nothing is known about the history of this problem except that Ferrari succeeded where Tartaglia and Cardan had failed.
Rheticus. Georg Joachim Rheticus, born at Feldkirch on Feb. 15, 1514, and died at Kaschau on Dec. 4, 1576, was professor at Wittenberg, and subsequently studied under Copernicus whose works were produced under the direction of Rheticus. Rheticus constructed various trigonometrical tables, some of which were published by his pupil Otho in 1596. These were subsequently completed and extended by Vieta and Pitiscus, and are the basis of those still in use. Reticus also found the values of sin 2theta and sin 3theta in terms of sin theta and cos theta , and was aware that trigonometrical ratios might be defined by means of the ratios of the sides of a right-angled triangle without introducing a circle.
I add here the names of some other celebrated mathematicians of about the same time, though their works are now of little value to any save antiquarians. Franciscus Maurolycus, born at Messina of Greek parents in 1494, and died in 1575, translated numerous Latin and Greek mathematical works, and discussed the conics regarded as sections of a cone: his works were published at Venice in 1575. Jean Borrel, born in 1492 and died at Grenoble in 1572, wrote an algebra, founded on that of Stifel; and a history of the quadrature of the circle: his works were published at Lyons in 1559. Wilhelm Xylander, born at Augsburg on Dec. 26, 1532, and died on Feb. 10, 1576, at Heidelberg, where since 1558 he had been professor, brought out an edition of the works of Psellus in 1556; an edition of Euclid's Elements in 1562; an edition of the Arithmetic of Diophantus in 1575; and some minor works which were collected and published in 1577. Frederigo Commandino, born at Urbino in 1509, and died there on Sept. 3, 1575, published a translation of the works of Archimedes in 1558; selections from Apollonius and Pappus in 1566; an edition of Euclid's Elements in 1572; and selections from Aristarchus, Ptolemy, Hero, and Pappus in 1574: all being accompanied by commentaries. Jacques Peletier, born at le Mans on July 25, 1517, and died at Paris in July 1582, wrote text-books on algebra and geometry: most of the results of Stifel and Cardan are included in the former. Adrian Romanus, born at Louvain on Sept. 29, 1561, and died on May 4, 1625, professor of mathematics and medicine at the university of Louvain, was the first to prove the usual formula for sin(A+B) . And lastly, Bartholomäus Pitiscus, born on Aug. 24, 1561, and died at Heidelberg, where he was professor of mathematics, on July 2, 1613, published his Trigonometry in 1599: this contains the expressions for sin (A ± B) and cos (A ± B) in terms of the trigonometrical ratios of A and B .
About this time also several text-books were produced which if they did not extend the boundaries of the subject systematized it. In particular I may mention those by Ramus and Bombelli.
Ramus. Peter Ramus was born at Cuth in Picardy in 1515, and was killed at Paris in the massacre of St. Bartholomew on Aug. 24, 1572. He was educated at the university of Paris, and on taking his degree he astonished and charmed the university with the brilliant declamation he delivered on the thesis that everything Aristotle had taught was false. He lectured--for it will be remembered that in early days there were no professors--first at le Mans, and afterwards at Paris; at the latter he founded the first chair of mathematics. Besides some works on philosophy he wrote treatises on arithmetic, algebra, geometry (founded on Euclid), astronomy (founded on the works of Copernicus), and physics, which were long regarded on the Continent as the standard text-books in these subjects. They are collected in an edition of his works published at Bâle in 1569.
Bombelli. Closely following the publication of Cardan's great work, Rafaello Bombelli published in 1572 an algebra which is a systematic exposition of the knowledge then current on the subject. In the preface he traces the history of the subject, and alludes to Diophantus who, in spite of the notice of Regiomontanus, was still unknown in Europe. He discusses radicals, real and complex. He also treats the theory of equations, and shews that in the irreducible case of a cubic equation the roots are all real; and he remarks that the problem to trisect a given angle is the same as that of the solution of a cubic equation. Finally he gave a large collection of problems.
Bombelli's work is noticeable for his use of symbols which indicate an approach to index notation. Following in the steps of Stifel, he introduced a symbol 1 for the unknown quantity, 2 for its square, 3 for its cube, and so on, and therefore wrote x^2+5x-4 as 1 2 p. 5 1 m. 4 Stevinus in 1586 employed 1, 2, 3, ... in a similar way; and suggested, though he did not use, a corresponding notation for fractional indices. He would have written the above expression as 1 2 + 5 1 - 4 0. But whether the symbols were more or less convenient they were still only abbreviations for words, and were subject to all the rules of syntax. They merely afforded a sort of shorthand by which the various steps and results could be expressed concisely. The next advance was the creation of symbolic algebra, and the chief credit of that is due to Vieta.
We have now reached a point beyond which any considerable development of algebra, so long as it was strictly syncopated, could hardly proceed. It is evident that Stifel and Bombelli and other writers of the sixteenth century had introduced or were on the point of introducing some of the ideas of symbolic algebra. But so far as the credit of inventing symbolic algebra can be put down to any one man we may perhaps assign it to Vieta, while we may say that Harriot and Descartes did more than any other writers to bring it into general use. It must be remembered, however, that it took time before all these innovations became generally known, and they were not familiar to mathematicians until the lapse of some years after they had been published.
Vieta.
Franciscus Vieta (François Viète) was born in 1540 at Fontenay near la Rochelle, and died in Paris in 1603. He was brought up as a lawyer and practised for some time at the Parisian bar; he then became a member of the provincial parliament in Brittany; and finally in 1580, through the influence of the Duke de Rohan, he was made master of requests, an office attached to the parliament at Paris; the rest of his life was spent in the public service. He was a firm believer in the right divine of kings, and probably a zealous catholic. After 1580 he gave up most of his leisure to mathematics, though his great work, In Artem Analyticam Isagoge, in which he explained how algebra could be applied to the solution of geometrical problems, was not published till 1591.
His mathematical reputation was already considerable, when one day the ambassador from the Low Countries remarked to Henry IV. that France did not possess any geometricians capable of solving a problem which had been propounded in 1593 by his countryman Adrian Romanus to all the mathematicians of the world, and which required the solution of an equation of the 45th degree. The king thereupon summoned Vieta, and informed him of the challenge. Vieta saw that the equation was satisfied by the chord of a circle (of unit radius) which subtends an angle 2pi/45 at the centre, and in a few minutes he gave back to the king two solutions of the problem written in pencil. In explanation of this feat I should add that Vieta had previously discovered how to form the equation connecting sin ntheta with sin theta and cos theta . Vieta in his turn asked Romanus to give a geometrical construction to describe a circle which should touch three given circles. This was the problem which Apollonius had treated in his De Tactionibus, a lost book which Vieta at a later time conjecturally restored. Romanus solved the problem by the use of conic sections, but failed to do it by Euclidean geometry. Vieta gave a Euclidean solution which so impressed Romanus that he travelled to Fontenay, where the French court was then settled, to make Vieta's acquaintance--an acquaintanceship which rapidly ripened into warm friendship.
Henry was much struck with the ability shown by Vieta in this matter. The Spaniards had at that time a cipher containing nearly 600 characters, which was periodically changed, and which they believed it was impossible to decipher. A despatch having been intercepted, the king gave it to Vieta, and asked him to try to read it and find the key to the system. Vieta succeeded, and for two years the French used it, greatly to their profit, in the war which was then raging. So convinced was Philip II. that the cipher could not be discovered, that when he found his plans known he complained to the Pope that the French were using sorcery against him, "contrary to the practice of the Christian faith."
Vieta wrote numerous works on algebra and geometry. The most important are the In Artem Analyticam Isagoge, Tours, 1591; the Supplementum Geometriae, and a collection of geometrical problems, Tours, 1593; and the De Numerosa Potestatum Resolutione, Paris, 1600. All of these were printed for private circulation only, but they were collected by F. van Schooten and published in one volume at Leyden in 1646. Vieta also wrote the De Aequationum Recognitione et Emendatione, which was published after his death in 1615 by Alexander Anderson.
The In Artem is the earliest work on symbolic algebra. It also introduced the use of letters for both known and unknown (positive) quantities, a notation for the powers of quantities, and enforced the advantage of working with homogeneous equations. To this an appendix called Logistice Speciosa was added on addition and multiplication of algebraical quantities, and on the powers of a binomial up to the sixth. Vieta implies that he knew how to form the coefficients of these six expansions by means of the arithmetical triangle as Tartaglia had previously done, but Pascal gave the general rule for forming it for any order, and Stifel had already indicated the method in the expansion of (1+x)^n if those in the expansion of (1+x)^n-1 were known; Newton was the first to give the general expression for the coefficient of x^p in the expansion of (1+x)^n . Another appendix known as Zetetica on the solution of equations was subsequently added to the In Artem.
The In Artem is memorable for two improvements in algebraic notation which were introduced here, though it is probable that Vieta took the idea of both from other authors.
One of these improvements was that he denoted the known quantities by the consonants B, C, D , &c., and the unknown quantities by the vowels A, E, I , &c. Thus in any problem he was able to use a number of unknown quantities. In this particular point he seems to have been forestalled by Jordanus and by Stifel. The present custom of using the letters at the beginning of the alphabet a, b, c , &c., to represent known quantities and those towards the end, x, y, z , &c., to represent the unknown quantities was introduced by Descartes in 1637.
The other improvement was this. Till this time it had been generally the custom to introduce new symbols to represent the square, cube, &c., of quantities which had already occurred in the equations; thus, if R or N stood for x , Z or C or Q stood for x^2 , and C or K for x^3 , &c. So long as this was the case the chief advantage of algebra was that it afforded a concise statement of results every statement of which was reasoned out. But when Vieta used A to denote the unknown quantity x , he sometimes employed A quadratus, A cubus, to represent x^2, x^3 , , which at once showed the connection between the different powers; and later the successive powers of A were commonly denoted by the abbreviations Aq, Ac, Aqq , &c. Thus Vieta would have written the equation 3BA^2-DA+A^3=Z, as B 3 in A quad. -- D plano in A + A cubo aequatur Z solido. It will be observed that the dimensions of the constants ( B, D, and Z ) are chosen so as to make the equation homogeneous: this is characteristic of all his work. It will be also noticed that he does not use a sign for equality; and in fact the particular sign = which we use to denote equality was employed by him to represent "the difference between." Vieta's notation is not so convenient as that previously used by Stifel, Bombelli, and Stevinus, but it was more generally adopted.
These two steps were almost essential to any further progress in algebra. In both of them Vieta had been forestalled, but it was his good luck in emphasising their importance to be the means of making them generally known at a time when opinion was ripe for such an advance.
The De Aequationum Recognitione et Emendatione is mostly on the theory of equations. It was not published till twelve years after Vieta's death, and it is possible that the editor made additions to it. Vieta here indicated how from a given equation another could be obtained whose roots were equal to those of the original increased by a given quantity, or multiplied by a given quantity; he used this method to get rid of the coefficient of x in a quadratic equation and of the coefficient of x^2 in a cubic equation, and was thus enabled to give the general algebraic solution of both. It would seem that he knew that the first member of an algebraical equation phi(x)=0 could be resolved into linear factors, and that the coefficients of x could be expressed as functions of the roots; perhaps the discovery of both these theorems should be attributed to him.
His solution of a cubic equation is as follows. First reduce the equation to the form x^3+3a^2x=2b^3 . Next let x=a^2/y-y , and we get y^6 + 2b^3y^3=a^6 , which is a quadratic in y^3 . Hence y can be found, and therefore x can be determined.
His solution of a biquadratic is similar to that known as Ferrari's, and essentially as follows. He first got rid of the term involving x^3 , thus reducing the equation to the form x^4 + a^2 x^2 + b^3x= c^4 . He then took the forms involving x^2 and x to the right-hand side of the equation and added x^2y^2+1/4y^4 to each side, so that the equation became (x^2+12y^2)^2=x^2(y^2-a^2)-b^3x+14y^4+c^4. He then chose y so that the right-hand side of this equality is a perfect square. Substituting this value of y , he was able to take the square root of both sides, and thus obtain two quadratic equations for x , each of which can be solved.
The De Numerosa Potestatum Resolutione deals with numerical equations. In this a method for approximating to the values of positive roots is given, but it is prolix and of little use, though the principle (which is similar to that of Newton's rule) is correct. Negative roots are uniformly rejected. This work is hardly worthy of Vieta's reputation.
Vieta's trigonometrical researches are included in various tracts which are collected in Van Schooten's edition. Besides some trigonometrical tables he gave the general expression for the sine (or chord) of an angle in terms of the sine and cosine of its submultiples. Delambre considers this as the completion of the Arab system of trigonometry. We may take it then that from this time the results of elementary trigonometry were familiar to mathematicians. Vieta also elaborated the theory of right-angled spherical triangles.
Among Vieta's miscellaneous tracts will be found a proof that each of the famous geometrical problems of the trisection of an angle and the duplication of the cube depends on the solution of a cubic equation. There are also some papers connected with an angry controversy with Clavius, in 1594, on the subject of the reformed calendar, in which Vieta was not well advised.
Vieta's works on geometry are good, but they contain nothing which requires mention here. He applied algebra and trigonometry to help him in investigating the properties of figures. He also, as I have already said, laid great stress on the desirability of always working with homogeneous equations, so that if a square or a cube were given it should be denoted by expressions like a^2 or b^3 , and not by terms like m or n which do not indicate the dimensions of the quantities they represent. He had a lively dispute with Scaliger on the latter publishing a solution of the quadrature of the circle, and Vieta succeeded in showing the mistake into which his rival had fallen. He gave a solution of his own which as far as it goes is correct, and stated that the area of a square is to that of the circumscribing circle as
12 × (12 + 12) × 12 + (12 + 12) ad inf. : 1.
This is one of the earliest attempts to find the value of pi by means of an infinite series. He was well acquainted with the extant writings of the Greek geometricians, and introduced the curious custom, which during the seventeenth and eighteenth centuries became fashionable, of restoring lost classical works. He himself produced a conjectural restoration of the De Tactionibus of Apollonius.
Girard. Vieta's results in trigonometry and the theory of equations were extended by Albert Girard, a Dutch mathematician, who was born in Lorraine in 1595, and died on December 9, 1632.
In 1626 Girard published at the Hague a short treatise on trigonometry, to which were appended tables of the values of the trigonometrical functions. This work contains the earliest use of the abbreviations sin, tan, sec for sine, tangent, and secant. The supplemental triangles in spherical trigonometry are also discussed; their properties seem to have been discovered by Girard and Snell at about the same time. Girard also gave the expression for the area of a spherical triangle in terms of the spherical excess--this was discovered independently by Cavalieri. In 1627 Girard brought out an edition of Marolois's Geometry with considerable additions.
Girard's algebraical investigations are contained in his Invention nouvelle en l'algèbre, published at Amsterdam in 1629. This contains the earliest use of brackets; a geometrical interpretation of the negative sign; the statement that the number of roots of an algebraical question is equal to its degree; the distinct recognition of imaginary roots; the theorem, known as Newton's rule, for finding the sum of like powers of the roots of an equation; and (in the opinion of some writers) implies also a knowledge that the first member of an algebraical equation phi(x) = 0 could be resolved into linear factors. Girard's investigations were unknown to most of his contemporaries, and exercised no appreciable influence on the development of mathematics.
The invention of logarithms by Napier of Merchiston in 1614, and their introduction into England by Briggs and others, have been already mentioned in chapter xi. A few words on these mathematicians may be here added.
Napier. John Napier was born at Merchiston in 1550, and died on April 4, 1617. He spent most of his time on the family estate near Edinburgh, and took an active part in the political and religious controversies of the day; the business of his life was to show that the Pope was Antichrist, but his favourite amusement was the study of mathematics and science.
As soon as the use of exponents became common in algebra the introduction of logarithms would naturally follow, but Napier reasoned out the result without the use of any symbolic notation to assist him, and the invention of logarithms was the result of the efforts of many years with a view to abbreviate the processes of multiplication and division. It is likely that Napier's attention may have been partly directed to the desirability of facilitating computations by the stupendous arithmetical efforts of some of his contemporaries, who seem to have taken a keen pleasure in surpassing one another in the extent to which they carried multiplications and divisions. The trigonometrical tables by Rheticus, which were published in 1596 and 1613, were calculated in a most laborious way: Vieta himself delighted in arithmetical calculations which must have taken days of hard work, and of which the results often served no useful purpose: L. van Ceulen (1539--1610) practically devoted his life to finding a numerical approximation to the value of pi , finally in 1610 obtaining it correct to 35 places of decimals: while, to cite one more instance, P. A. Cataldi (1548--1626), who is chiefly known for his invention in 1613 of the form of continued fractions, must have spent years in numerical calculations.
In regard to Napier's other work I may again mention that in his Rabdologia, published in 1617, he introduced an improved form of rod by the use of which the product of two numbers can be found in a mechanical way, or the quotient of one number by another. He also invented two other rods called "virgulae," by which square and cube roots can be extracted. I should add that in spherical trigonometry he discovered certain formulae known as Napier's analogies, and enunciated the "rule of circular parts" for the solution of right-angled spherical triangles.
Briggs. The name of Briggs is inseparably associated with the history of logarithms. Henry Briggs was born near Halifax in 1561: he was educated at St. John's College, Cambridge, took his degree in 1581, and obtained a fellowship in 1588: he was elected to the Gresham professorship of geometry in 1596, and in 1619 or 1620 became Savilian professor at Oxford, a chair which he held until his death on January 26, 1631. It may be interesting to add that the chair of geometry founded by Sir Thomas Gresham was the earliest professorship of mathematics established in Great Britain. Some twenty years earlier Sir Henry Savile had given at Oxford open lectures on Greek geometry and geometricians, and in 1619 he endowed the chairs of geometry and astronomy in that university which are still associated with his name. Both in London and at Oxford Briggs was the first occupant of the chair of geometry. He began his lectures at Oxford with the ninth proposition of the first book of Euclid--that being the furthest point to which Savile had been able to carry his audiences. At Cambridge the Lucasian chair was established in 1663, the earliest occupants being Barrow and Newton.
The almost immediate adoption throughout Europe of logarithms for astronomical and other calculations was mainly the work of Briggs, who undertook the tedious work of calculating and preparing tables of logarithms. Amongst others he convinced Kepler of the advantages of Napier's discovery, and the spread of the use of logarithms was rendered more rapid by the zeal and reputation of Kepler, who by his tables of 1625 and 1629 brought them into vogue in Germany, while Cavalieri in 1624 and Edmund Wingate in 1626 did a similar service for Italian and French mathematicians respectively. Briggs also was instrumental in bringing into common use the method of long division now generally employed.
Harriot. Thomas Harriot, who was born at Oxford in 1560, and died in London on July 2, 1621, did a great deal to extend and codify the theory of equations. The early part of his life was spent in America with Sir Walter Raleigh; while there he made the first survey of Virginia and North Carolina, the maps of these being subsequently presented to Queen Elizabeth. On his return to England he settled in London, and gave up most of his time to mathematical studies.
The majority of the propositions I have assigned to Vieta are to be found in Harriot's writings, but it is uncertain whether they were discovered by him independently of Vieta or not. In any case it is probable that Vieta had not fully realised all that was contained in the propositions he had enunciated. Some of the consequences of these, with extensions and a systematic exposition of the theory of equations, were given by Harriot in his Artis Analyticae Praxis, which was first printed in 1631. The Praxis is more analytical than any algebra that preceded it, and marks an advance both in symbolism and notation, though negative and imaginary roots are rejected. It was widely read, and proved one of the most powerful instruments in bringing analytical methods into general use. Harriot was the first to use the signs > and < to represent greater than and less than. When he denoted the unknown quantity by a he represented a^2 by aa , a^3 by aaa , and so on. This is a distinct improvement on Vieta's notation. The same symbolism was used by Wallis as late as 1685, but concurrently with the modern index notation which was introduced by Descartes. I need not allude to the other investigations of Harriot, as they are comparatively of small importance; extracts from some of them were published by S. P. Rigaud in 1833.
Oughtred. Among those who contributed to the general adoption in England of these various improvements and additions to algorism and algebra was William Oughtred, who was born at Eton on March 5, 1574, and died at his vicarage of Albury in Surrey on June 30, 1660: it is sometimes said that the cause of his death was the excitement and delight which he experienced "at hearing the House of Commons [or Convention] had voted the King's return"; a recent critic adds that it should be remembered "by way of excuse that he [Oughtred] was then eighty-six years old," but perhaps the story is sufficiently discredited by the date of his death. Oughtred was educated at Eton and King's College, Cambridge, of the latter of which colleges he was a fellow and for some time mathematical lecturer.
His Clavis Mathematicae published in 1631 is a good systematic text-book on arithmetic, and it contains practically all that was then known on the subject. In this work he introduced the symbol × for multiplication. He also introduced the symbol nbproportion in proportion: previously to his time a proportion such as a : b = c : d was usually written as a-b-c-d ; he denoted it by a ctrdot b nbproportion c ctrdot d . Wallis says that some found fault with the book on account of the style, but that they only displayed their own incompetence, for Oughtred's "words be always full but not redundant." Pell makes a somewhat similar remark.
Oughtred also wrote a treatise on trigonometry published in 1657, in which abbreviations for sine, cosine, &c., were employed. This was really an important advance, but the works of Girard and Oughtred, in which they were used, were neglected and soon forgotten, and it was not until Euler reintroduced contractions for the trigonometrical functions that they were generally adopted. In this work the colon (i.e. the symbol : ) was used to denote a ratio.
We may say roughly that henceforth elementary arithmetic, algebra, and trigonometry were treated in a manner which is not substantially different from that now in use; and that the subsequent improvements introduced were additions to the subjects as then known, and not a rearrangement of them on new foundations.
It may be convenient if I collect here in one place the scattered remarks I have made on the introduction of the various symbols for the more common operations in algebra.
, 1879, vol. xxxvii, pp. 324--333, vol. xxxviii, pp. 1--10.
The later Greeks, the Hindoos, and Jordanus indicated addition by mere juxtaposition. It will be observed that this is still the custom in arithmetic, where, for instance, 21/2 stands for 2 + 1/2 . The Italian algebraists, when they gave up expressing every operation in words at full length and introduced syncopated algebra, usually denoted plus by its initial letter P or p , a line being sometimes drawn through the letter to show that it was a contraction, or a symbol of operation, and not a quantity. The practice, however, was not uniform; Pacioli, for example, sometimes denoted plus by p , and sometimes by e , and Tartaglia commonly denoted it by phi . The German and English algebraists, on the other hand, introduced the sign + almost as soon as they used algorism, but they spoke of it as signum additorum and employed it only to indicate excess; they also used it with a special meaning in solutions by the method of false assumption. Widman used it as an abbreviation for excess in 1489: by 1630 it was part of the recognised notation of algebra, and was used as a symbol of operation.
Subtraction was indicated by Diophantus by an inverted and truncated psi . The Hindoos denoted it by a dot. The Italian algebraists when they introduced syncopated algebra generally denoted minus by M or m , a line being sometimes drawn through the letter; but the practice was not uniform--Pacioli, for example, denoting it sometimes by m , and sometimes by de for demptus. The German and English algebraists introduced the present symbol which they described as signum subtractorum. It is most likely that the vertical bar in the symbol for plus was superimposed on the symbol for minus to distinguish the two. It may be noticed that Pacioli and Tartaglia found the sign - already used to denote a division, a ratio, or a proportion indifferently. The present sign for minus was in general use by about the year 1630, and was then employed as a symbol of operation.
Vieta, Schooten, and others among their contemporaries employed the sign = written between two quantities to denote the difference between them; thus a = b means with them what we denote by a nbdiffc b . On the other hand, Barrow wrote nbdiffm for the same purpose. I am not aware when or by whom the current symbol nbdiffc was first used with this signification.
Oughtred in 1631 used the sign × to indicate multiplication; Harriot in 1631 denoted the operation by a dot; Descartes in 1637 indicated it by juxtaposition. I am not aware of any symbols for it which were in previous use. Leibnitz in 1686 employed the sign frown to denote multiplication.
Division was ordinarily denoted by the Arab way of writing the quantities in the form of a fraction by means of a line drawn between them in any of the forms a - b , a/b , or a/b . Oughtred in 1631 employed a dot to denote either division or a ratio. Leibnitz in 1686 employed the sign smile to denote division. The colon (or symbol : ), used to denote a ratio, occurs on the last two pages of Oughtred's Canones Sinuum, published in 1657. I believe that the current symbol for division ÷ is only a combination of the - and the symbol : for a ratio; it was used by Johann Heinrich Rahn at Zürich in 1659, and by John Pell in London in 1668. The symbol nbbarrowdiv was used by Barrow and other writers of his time to indicate continued proportion.
The current symbol for equality was introduced by Record in 1557; Xylander in 1575 denoted it by two parallel vertical lines; but in general till the year 1600 the word was written at length; and from then until the time of Newton, say about 1680, it was more frequently represented by nbinftyleft or by nbinftyright than by any other symbol. Either of these latter signs was used as a contraction for the first two letters of the word aequalis.
The symbol nbproportion to denote proportion, or the equality of two ratios, was introduced by Oughtred in 1631, and was brought into common use by Wallis in 1686. There is no object in having a symbol to indicate the equality of two ratios which is different from that used to indicate the equality of other things, and it is better to replace it by the sign = .
The sign > for is greater than and the sign < for is less than were introduced by Harriot in 1631, but Oughtred simultaneously invented the symbols nboughtredgreater and nboughtredless for the same purpose; and these latter were frequently used till the beginning of the eighteenth century, ex. gr. by Barrow.
The symbols nbne for is not equal to, nbngt for is not greater than, and nbnlt for is not less than, are, I believe, now rarely used outside Great Britain; they were employed, if not invented, by Euler. The symbols geqq and leqq were introduced by P. Bouguer in 1734.
The vinculum was introduced by Vieta in 1591; and brackets were first used by Girard in 1629.
The symbol sqrt to denote the square root was introduced by Rudolff in 1526; a similar notation had been used by Bhaskara and by Chuquet.
The different methods of representing the power to which a magnitude was raised have been already briefly alluded to. The earliest known attempt to frame a symbolic notation was made by Bombelli in 1572, when he represented the unknown quantity by 1, its square by 2, its cube by 3, &c. In 1586 Stevinus used 1, 2, 3, &c., in a similar way; and suggested, though he did not use, a corresponding notation for fractional indices. In 1591 Vieta improved on this by denoting the different powers of A by A, A quad., A cub., &c., so that he could indicate the powers of different magnitudes; Harriot in 1631 further improved on Vieta's notation by writing aa for a^2 , aaa for a^3 , &c., and this remained in use for fifty years concurrently with the index notation. In 1634 P. Herigonus, in his Cursus mathematicus, published in five volumes at Paris in 1634--1637, wrote a, a2, a3, dots for a, a^2, a^3 dots .
The idea of using exponents to mark the power to which a quantity was raised was due to Descartes, and was introduced by him in 1637; but he used only positive integral indices a^1, a^2, a^3, . Wallis in 1659 explained the meaning of negative and fractional indices in expressions such as a^-1 , ax^1/2 , &c.; the latter conception having been foreshadowed by Oresmus and perhaps by Stevinus. Finally the idea of an index unrestricted in magnitude, such as the n in the expression a^n , is, I believe, due to Newton, and was introduced by him in connection with the binomial theorem in the letters for Leibnitz written in 1676.
The symbol infinity for infinity was first employed by Wallis in 1655 in his Arithmetica Infinitorum; but does not occur again until 1713, when it is used in James Bernoulli's Ars Conjectandi. This sign was sometimes employed by the Romans to denote the number 1000, and it has been conjectured that this led to its being applied to represent any very large number.
There are but few special symbols in trigonometry; I may, however, add here the following note which contains all that I have been able to learn on the subject. The current sexagesimal division of angles is derived from the Babylonians through the Greeks. The Babylonian unit angle was the angle of an equilateral triangle; following their usual practice this was divided into sixty equal parts or degrees, a degree was subdivided into sixty equal parts or minutes, and so on; it is said that 60 was assumed as the base of the system in order that the number of degrees corresponding to the circumference of a circle should be the same as the number of days in a year which it is alleged was taken (at any rate in practice) to be 360.
The word sine was used by Regiomontanus and was derived from the Arabs; the terms secant and tangent were introduced by Thomas Finck (born in Denmark in 1561 and died in 1646) in his Geometriae Rotundi, Bâle, 1583; the word cosecant was (I believe) first used by Rheticus in his Opus Palatinum, 1596; the terms cosine and cotangent were first employed by E. Gunter in his Canon Triangulorum, London, 1620. The abbreviations sin, tan, sec were used in 1626 by Girard, and those of cos and cot by Oughtred in 1657; but these contractions did not come into general use till Euler reintroduced them in 1748. The idea of trigonometrical functions originated with John Bernoulli, and this view of the subject was elaborated in 1748 by Euler in his Introductio in Analysin Infinitorum.
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End of part 16 of 28
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