A History of Mathematics/Modern Europe/The Renaissance

by Florian Cajori

 

THE RENAISSANCE.

 

With the sixteenth century began a period of increased intellectual activity. The human mind made a vast effort to achieve its freedom. Attempts at its emancipation from Church authority had been made before, but they were stifled and rendered abortive. The first great and successful revolt against ecclesiastical authority was made in Germany. The new desire for judging freely and independently in matters of religion was preceded and accompanied by a growing spirit of scientific inquiry. Thus it was that, for a time, Germany led the van in science. She produced Regiomontanus, Copernicus, Rhœticus, Kepler, and Tycho Brahe, at a period when France and England had, as yet, brought forth hardly any great scientific thinkers. This remarkable scientific productiveness was no doubt due, to a great extent, to the commercial prosperity of Germany. Material prosperity is an essential condition for the progress of knowledge. As long as every individual is obliged to collect the necessaries for his subsistence, there can be no leisure for higher pursuits. At this time, Germany had accumulated considerable wealth. The Hanseatic League commanded the trade of the North. Close commercial relations existed between Germany and Italy. Italy, too, excelled in commercial activity and enterprise. We need only mention Venice, whose glory began with the crusades, and Florence, with her bankers and her manufacturers of silk and wool. These two cities became great intellectual centres. Thus, Italy, too, produced men in art, literature, and science, who shone forth in fullest splendour. In fact, Italy was the fatherland of what is termed the Renaissance.

For the first great contributions to the mathematical sciences we must, therefore, look to Italy and Germany. In Italy brilliant accessions were made to algebra, in Germany to astronomy and trigonometry.

On the threshold of this new era we meet in Germany with the figure of John Mueller, more generally called Regiomontanus (1436-1476). Chiefly to him we owe the revival of trigonometry. He studied astronomy and trigonometry at Vienna under the celebrated George Purbach. The latter perceived that the existing Latin translations of the Almagest were full of errors, and that Arabic authors had not remained true to the Greek original. Purbach therefore began to make a translation directly from the Greek. But he did not live to finish it. His work was continued by Regiomontanus, who went beyond his master. Regiomontanus learned the Greek language from Cardinal Bessarion, whom he followed to Italy, where he remained eight years collecting manuscripts from Greeks who had fled thither from the Turks. In addition to the translation of and the commentary on the Almagest, he prepared translations of the Conics of Apollonius, of Archimedes, and of the mechanical works of Heron. Regiomontanus and Purbach adopted the Hindoo sine in place of the Greek chord of double the arc. The Greeks and afterwards the Arabs divided the radius into 60 equal parts, and each of these again into 60 smaller ones. The Hindoos expressed the length of the radius by parts of the circumference, saying that of the 21,600 equal divisions of the latter, it took 3438 to measure the radius. Regiomontanus, to secure greater precision, constructed one table of sines on a radius divided into 600,000 parts, and another on a radius divided decimally into 10,000,000 divisions. He emphasised the use of the tangent in trigonometry. Following out some ideas of his master, he calculated a table of tangents. German mathematicians were not the first Europeans to use this function. In England it was known a century earlier to Bradwardine, who speaks of tangent (umbra recta) and cotangent (umbra versa), and to John Maudith. Regiomontanus was the author of an arithmetic and also of a complete treatise on trigonometry, containing solutions of both plane and spherical triangles. The form which he gave to trigonometry has been retained, in its main features, to the present day.

Regiomontanus ranks among the greatest men that Germany has ever produced. His complete mastery of astronomy and mathematics, and his enthusiasm for them, were of far-reaching influence throughout Germany. So great was his reputation, that Pope Sixtus IV, called him to Italy to improve the calendar. Regiomontanus left his beloved city of Nürnberg for Rome, where he died in the following year.

After the time of Purbach and Regiomontanus, trigonometry and especially the calculation of tables continued to occupy German scholars. More refined astronomical instruments were made, which gave observations of greater precision; but these would have been useless without trigonometrical tables of corresponding accuracy. Of the several tables calculated, that by Georg Joachim of Feldkirch in Tyrol, generally called Rhæticus, deserves special mention. He calculated a table of sines with the radius = 10,000,000,000 and from 10" to 10"; and, later on, another with the radius = 1,000,000,000,000,000, and proceeding from 10" to 10". He began also the construction of tables of tangents and secants, to be carried to the same degree of accuracy; but he died before finishing them. For twelve years he had had in continual employment several calculators. The work was completed by his pupil, Valentine Otho; in 1596. This was indeed a gigantic work,—a monument of German diligence and indefatigable perseverance. The tables were republished in 1613 by Pitiscus, who spared no pains to free them of errors. Astronomical tables of so great a degree of accuracy had never been dreamed of by the Greeks, Hindoos, or Arabs. That Rhæticus was not a ready calculator only, is indicated by his views on trignometrical lines. Up to his time, the trigonometric functions had been considered always with relation to the arc; he was the first to construct the right triangle and to make them depend directly upon its angles. It was from the right triangle that Rhæticus go this idea of calculating the hypotenuse; i.e. he was the first to plan a table of secants. Good work in trigonometry was done also by Vieta and Romanus.

We shall now leave the subject of trigonometry to witness the progress in the solution of algebraical equations. To do so, we must quit Germany for Italy. The first comprehensive algebra printed was that of Lucas Pacioli. He closes his book by saying that the solution of the equations $\scriptstyle {x^{3}+mx=n,~x^{3}+n=mx}$ is as impossible at the present state of science as the quadrature of the circle. This remark doubtless stimulated thought. The first step in the algebraic solution of cubics was taken by Scipio Ferro (died 1526), a professor of mathematics at Bologna, who solved the equation $\scriptstyle {x^{3}+mx=n}$. Nothing more is known of his discovery than that he imparted it to his pupil, Floridas, in 1505. It was the practice in those days and for two centuries afterwards to keep discoveries secret, in order to secure by that means an advantage over rivals by proposing problems beyond their reach. This practice gave rise to numberless disputes regarding the priority of inventions. A second solution of cubics was given by Nicolo of Brescia (1506(?)-1657). When a boy of six, Nicolo was so badly cut by a French soldier that he never again gained the free use of his tongue. Hence he was called Tartaglia, i.e. the stammerer. His widowed mother being too poor to pay his tuition in school, he learned to read and picked up a knowledge of Latin, Greek, and mathematics by himself. Possessing a mind of extraordinary power, he was able to appear as teacher of mathematics at an early age. In 1530, one Colla proposed him several problems, one leading to the equation $\scriptstyle {x^{3}+px^{2}=q}$. Tartaglia found an imperfect method for solving this, but kept it secret. He spoke about his secret in public and thus caused Ferro's pupil, Floridas, to proclaim his own knowledge of the form $\scriptstyle {x^{3}+mx=n}$. Tartaglia, believing him to be a mediocrist and braggart, challenged him to a public discussion, to take place on the 22d of February, 1535. Hearing, meanwhile, that his rival had gotten the method from a deceased master, and fearing that he would be beaten in the contest, Tartaglia put in all the zeal, industry, and skill to find the rule for the equations, and he succeeded in it ten days before the appointed date, as he himself modestly says.^[7] The most difficult step was, no doubt, the passing from quadratic irrationals, used in operating from time of old, to cubic irrationals. Placing $\scriptstyle {x={\sqrt[{3}]{t}}-{\sqrt[{3}]{u}}}$, Tartaglia perceived that the irrationals disappeared from the equation $\scriptstyle {x^{3}+mx=n}$, making $\scriptstyle {n=t-u}$. But this last equality, together with $\scriptstyle {({\frac {1}{3}}m)^{3}=tu}$, gives at once

$\scriptstyle {t={\sqrt {({\frac {n}{2}})^{3}+({\frac {m}{3}})^{3}}}+{\frac {n}{2}},~u={\sqrt {({\frac {n}{2}})^{2}+({\frac {m}{2}})^{3}}}-{\frac {n}{2}}}$.

This is Tartaglia's solution of $\scriptstyle {x^{3}+mx=n}$. On the 13th of February, he found a similar solution for $\scriptstyle {x^{3}=mx+n}$. The contest began on the 22d. Each contestant proposed thirty problems. The one who could solve the greatest number within fifty days should be the victor. Tartaglia solved the thirty problems proposed by Floridas in two hours; Floridas could not solve any of Tartaglia's. From now on, Tartaglia studied cubic equations with a will. In 1541 he discovered a general solution for the cubic $\scriptstyle {x^{3}\pm px^{2}=\pm q}$, by transforming it into the form $\scriptstyle {x^{3}\pm mx=\pm n}$. The news of Tartaglia's victory spread all over Italy. Tartaglia was entreated to make known his method, but he declined to do so, saying that after his completion of the translation from the Greek of Euclid and Archimedes, he would publish a large algebra containing his method. But a scholar from Milan, named Hieronimo Cardano (1501-1576), after many solicitations, and after giving the most solemn and sacred promises of secrecy, succeeded in obtaining from Tartaglia a knowledge of his rules.

At this time Cardan was writing his Ars Magna, and he knew no better way to crown his work than by inserting the much sought for rules for solving cubics. Thus Cardan broke his most solemn vows, and published in 1545 in his Ars Magna Tartaglia's solution of cubics. Tartaglia became desperate. His most cherished hope, of giving to the world an immortal work which should be the monument of his deep learning and power for original research, was suddenly destroyed; for the crown intended for his work had been snatched away. His first step was to write a history of his invention; but, to completely annihilate his enemies, he challenged Cardan and his pupil Lodovico Ferrari to a contest: each party should propose thirty-one questions to be solved by the other within fifteen days. Tartaglia solved most questions in seven days, but the other party did not send in their solution before the expiration of the fifth month; moreover, all their solutions except one were wrong. A replication and a rejoinder followed. Endless were the problems proposed and solved on both sides. The dispute produced much chagrin and heart-burnings to the parties, and to Tartaglia especially, who met with many other disappointments. After having recovered himself again, Tartaglia began, in 1556, the publication of the work which he had had in his mind for so long; but he died before he reached the consideration of cubic equations. Thus the fondest wish of his life remained unfulfilled; the man to whom we owe the greatest contribution to algebra made in the sixteenth century was forgotten, and his method came to be regarded as the discovery of Cardan and to be called Cardan's solution.

Remarkable is the great interest that the solution of cubics excited throughout Italy. It is but natural that after this great conquest mathematicians should attack bi-quadratic equations. As in the case of cubics, so here, the first impulse was given by Colla, who, in 1540, proposed for solution the equation $\scriptstyle {x^{4}+6x^{2}+36=60x}$. To be sure. Cardan had studied particular cases as early as 1539. Thus he solved the equation $\scriptstyle {13x^{2}=x^{4}+2x^{3}+2x+1}$ by a process similar to that employed by Diophantus and the Hindoos; namely, by adding to both sides $\scriptstyle {3x^{2}}$ and thereby rendering both numbers complete squares. But Cardan failed to find a general solution; it remained for his pupil Ferrari to prop the reputation of his master by the brilliant discovery of the general solution of bi-quadratic equations. Ferrari reduced Colla's equation to the form $\scriptstyle {(x^{2}+6)^{2}=60x+6x^{2}}$. In order to give also the right member the form of a complete square he added to both members the expression $\scriptstyle {2(x^{2}+6)y+y^{2}}$, containing a new unknown quantity y. This gave him $\scriptstyle {(x^{2}+6+y)^{2}=(6+2y)x^{2}+60x+(12y+y^{2})}$. The condition that the right member be a complete square is expressed by the cubic equation $\scriptstyle {(2y+6)(12y+y^{2})=900}$. Extracting the square root of the bi-quadratic, he got $\scriptstyle {x^{2}+6+y=x{\sqrt {2y+6}}+{\frac {900}{\sqrt {2y+6}}}}$. Solving the cubic for y and substituting, it remained only to determine x from the resulting quadratic. Ferrari pursued a similar method with other numerical bi-quadratic equations.^[7] Cardan had the pleasure of publishing this discovery in his Ars Magna in 1545. Ferrari's solution is sometimes ascribed to Bombelli, but he is no more the discoverer of it than Cardan is of the solution called by his name.

To Cardan algebra is much indebted. In his Ars Magna he takes notice of negative roots of an equation, calling them fictitious, while the positive roots are called real. Imaginary roots he does not consider; cases where they appear he calls impossible. Cardan also observed the difficulty in the irreducible case in the cubics, which, like the quadrature of the circle, has since "so much tormented the perverse ingenuity of mathematicians." But he did not understand its nature. It remained for Raphael Bombelli of Bologna, who published in 1572 an algebra of great merit, to point out the reality of the apparently imaginary expression which the root assumes, and thus to lay the foundation of a more intimate knowledge of imaginary quantities.

After this brilliant success in solving equations of the third and fourth degrees, there was probably no one who doubted, that with aid of irrationals of higher degrees, the solution of equations of any degree whatever could be found. But all attempts at the algebraic solution of the quintic were fruitless, and, finally, Abel demonstrated that all hopes of finding algebraic solutions to equations of higher than the fourth degree were purely Utopian.

Since no solution by radicals of equations of higher degrees could be found, there remained nothing else to be done than the devising of rules by which at least the numerical values of the roots could be ascertained. Cardan applied the Hindoo rule of "false position" (called by him regula aurea) to the cubic, but this mode of approximating was exceedingly rough. An incomparably better method was invented by Franciscus Vieta, a French mathematician, whose transcendent genius enriched mathematics with several important innovations. Taking the equation $\scriptstyle {f(x)=Q}$, wherein $\scriptstyle {f(x)}$ is a polynomial containing different powers of x, with numerical coefficients, and Q is a given number, Vieta first substitutes in $\scriptstyle {f(x)}$ a known approximate value of the root, and then shows that another figure of the root can be obtained by division. A repetition of the same process gives the next figure of the root, and so on. Thus, in $\scriptstyle {x^{2}+14x=7929}$, taking 80 for the approximate root, and placing $\scriptstyle {x=80+b}$, we get

or

${\begin{aligned}&\scriptstyle {(80+b)^{2}+14(80+b)=7929},\\&\scriptstyle {174b+b^{2}=409}\end{aligned}}$.

Since $\scriptstyle {174b}$ is much greater than $\scriptstyle {b^{2}}$, we place $\scriptstyle {174b=409}$, and obtain thereby $\scriptstyle {b=2}$. Hence the second approximation is 82. Put $\scriptstyle {x=82+c}$, then $\scriptstyle {(82+c)^{2}+14(82+c)=7929}$, or $\scriptstyle {178c+c^{2}=57}$. As before, place $\scriptstyle {178c=57}$, then $\scriptstyle {c=.3}$, and the third approximation gives 82.3. Assuming $\scriptstyle {x=82.3+d}$, and substituting, gives $\scriptstyle {178.6d+d^{2}=3.51}$, and $\scriptstyle {178.6d=3.51}$, $\scriptstyle {\therefore ~d=.01}$; giving for the fourth approximation 82.31. In the same way, $\scriptstyle {e=.009}$, and the value for the root of the given equation is 82.319… For this process, Vieta was greatly admired by his contemporaries. It was employed by Harriot, Oughtred, Pell, and others. Its principle is identical with the main principle involved in the methods of approximation of Newton and Homer. The only change lies in the arrangement of the work. This alteration was made to afford facility and security in the process of evolution of the root.

We pause a moment to sketch the life of Vieta, the most eminent French mathematician of the sixteenth century. He was born in Poitou in 1540, and died in 1603 at Paris. He was employed throughout life in the service of the state, under Henry III. and Henry IV. He was, therefore, not a mathematician by profession, but his love for the science was so great that he remained in his chamber studying, sometimes several days in succession, without eating and sleeping more than was necessary to sustain himself. So great devotion to abstract science is the more remarkable, because he lived at a time of incessant political and religious turmoil. During the war against Spain, Vieta rendered service to Henry IV. by deciphering intercepted letters written in a species of cipher, and addressed by the Spanish Court to their governor of Netherlands. The Spaniards attributed the discovery of the key to magic.

An ambassador from Netherlands once told Henry IV. that France did not possess a single geometer capable of solving a problem propounded to geometers by a Belgian mathematician, Adrianus Romanus. It was the solution of the equation of the forty-fifth degree:—

$\scriptstyle {45y-3795y^{2}+95634y^{3}-\cdots +945y^{41}-45y^{41}+y^{41}=C}$.

Henry IV. called Vieta, who, having already pursued similar investigations, saw at once that this awe-inspiring problem was simply the equation by which $\scriptstyle {C=2\sin \phi }$ was expressed in terms of $\scriptstyle {y=2\sin {\frac {1}{45}}\phi }$; that, cince $\scriptstyle {45=3\cdot 3\cdot 5}$, it was necessary only to divide an angle once into 5 equal parts, and then twice into 3,—a division which could be effected by corresponding equations of the fifth and third degrees. Brilliant was the discovery by Vieta of 33 roots to this equation, instead of only one. The reason why he did not find 45 solutions, is that the remaining ones involve negative sines, which were unintelligible to him. Detailed investigations on the famous old problem of the section of an angle into an odd number of equal parts, led Vieta to the discovery of a trigonometrical solution of Cardan's irreducible case in cubics. He applied the equation $\scriptstyle {(2\cos {\frac {1}{3}}\phi )^{3}-3(2\cos {\frac {1}{3}}\phi )=2\cos \phi }$ to the solution of $\scriptstyle {x^{3}-3a^{2}x=a^{2}b}$, when $\scriptstyle {a>{\frac {1}{2}}b}$, by placing $\scriptstyle {x=2a\cos {\frac {1}{3}}\phi }$, and determining $\scriptstyle {\phi }$ from $\scriptstyle {b=2a\cos \phi }$.

The main principle employed by him in the solution of equations is that of reduction. He solves the quadratic by making a suitable substitution which will remove the term containing x to the first degree. Like Cardan, he reduces the general expression of the cubic to the form $\scriptstyle {x^{3}+mx+n=0}$; then, assuming $\scriptstyle {x=({\frac {1}{3}}a-z^{2})\div z}$ and substituting, he gets $\scriptstyle {z^{6}-bz^{3}-{\frac {1}{27}}a^{3}=0}$. Putting $\scriptstyle {z^{3}=y}$, he has a quadratic. In the solution of bi-quadratics, Vieta still remains true to his principle of reduction. This gives him the well-known cubic resolvent. He thus adheres throughout to his favourite principle, and thereby introduces into algebra a uniformity of method which claims our lively admiration. In Vieta's algebra we discover a partial knowledge of the relations existing between the coefficients and the roots of an equation. He shows that if the coefficient of the second term in an equation of the second degree is minus the sum of two numbers whose product is the third term, then the two numbers are roots of the equation. Vieta rejected all except positive roots; hence it was impossible for him to fully perceive the relations in question.

The most epoch-making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him logistica speciosa in distinction to the old logistica numerosa. Vieta's formalism differed considerably from that of to-day. The equation $\scriptstyle {a^{3}+3a^{2}b+3ab^{2}+b^{3}=(a+b)^{3}}$ was written by him "a cubus + b in a quadr. 3 + a in b quadr. 3 + b cubo æqualia a + b cubo." In numerical equations the unknown quantity was denoted by N, its square by Q, and its cube by C. Thus the equation $\scriptstyle {x^{3}-8x^{2}+16x=40}$ was written $\scriptstyle {1C-8Q+16N}$ æqual. 40. Observe that exponents and our symbol ( = ) for equality were not yet in use; but that Vieta employed the Maltese cross ( + ) as the short-hand symbol for addition, and the ( — ) for subtraction. These two characters had not been in general use before the time of Vieta. "It is very singular," says Hallam, "that discoveries of the greatest convenience, and, apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari; and hardly less so that, by dint of that acuteness, they dispensed with the aid of these contrivances in which we suppose that so much of the utility of algebraic expression consists." Even after improvements in notation were once proposed, it was with extreme slowness that they were admitted into general use. They were made oftener by accident than design, and their authors had little notion of the effect of the change which they were making. The introduction of the + and — symbols seems to be due to the Germans, who, although they did not enrich algebra during the Renaissance with great inventions, as did the Italians, still cultivated it with great zeal. The arithmetic of John Widmann, printed A.D. 1489 in Leipzig, is the earliest book in which the + and — symbols have been found. There are indications leading us to surmise that they were in use first among merchants. They occur again in the arithmetic of Grammateus, a teacher at the University of Vienna. His pupil, Christoff Rudolff, the writer of the first text-book on algebra in the German language (printed in 1525), employs these symbols also. So did Stifel, who brought out a second edition of Rudolff's Coss in 1553. Thus, by slow degrees, their adoption became universal. There is another short-hand symbol of which we owe the origin to the Germans. In a manuscript published sometime in the fifteenth century, a dot placed before a number is made to signify the extraction of a root of that number. This dot is the embryo of our present symbol for the square root. Christoff Rudolff, in his algebra, remarks that "the radix quadrata is, for brevity, designated in his algorithm with the character $\scriptstyle {\surd }$, as $\scriptstyle {\surd 4}$." Here the dot has grown into a symbol much like our own. This same symbol was used by Michael Stifel. Our sign of equality is due to Robert Recorde (1510-1558), the author of The Whetstone of Witte (1557), which is the first English treatise on algebra. He selected this symbol because no two things could be more equal than two parallel lines =. The sign $\scriptstyle {\div }$ for division was first used by Johann Heinrich Rahn, a Swiss, in 1659, and was introduced in England by John Pell in 1668.

Michael Stifel (1486?-1567), the greatest German algebraist of the sixteenth century, was born in Esslingen, and died in Jena. He was educated in the monastery of his native place, and afterwards became Protestant minister. The study of the significance of mystic numbers in Revelation and in Daniel drew him to mathematics. He studied German and Italian works, and published in 1544, in Latin, a book entitled Arithmetica integra, Melanchthon wrote a preface to it. Its three parts treat respectively of rational numbers, irrational numbers, and algebra. Stifel gives a table containing the numerical values of the binomial coefficients for powers below the 18th. He observes an advantage in letting a geometric progression correspond to an arithmetical progression, and arrives at the designation of integral powers by numbers. Here are the germs of the theory of exponents. In 1545 Stifel published an arithmetic in German. His edition of Rudolff's Coss contains rules for solving cubic equations, derived from the the writings of Cardan.

We remarked above that Vieta discarded negative roots of equations. Indeed, we find few algebraists before and during the Renaissance who understood the significance even of negative quantities. Fibonacci seldom uses them. Pacioli states the rule that "minus times minus gives plus," but applies it really only to the development of the product of $\scriptstyle {(a-b)(c-d)}$; purely negative quantities do not appear in his work. The great German "Cossist" (algebraist), Michael Stifel, speaks as early as 1544 of numbers which are "absurd" or "fictitious below zero," and which arise when "real numbers above zero," are subtracted from zero. Cardan, at last, speaks of a "pure minus"; "but these ideas," says Hankel, "remained sparsely, and until the beginning of the seventeenth century, mathematicians dealt exclusively with absolute positive quantities." The first algebraist who occasionally places a purely negative quantity by itself on one side of an equation, is Harriot in England. As regards the recognition of negative roots. Cardan and Bombelli were far in advance of all writers of the Renaissance, including Vieta. Yet even they mentioned these so-called false or fictitious roots only in passing, and without grasping their real significance and importance. On this subject Cardan and Bombelli had advanced to about the same point as had the Hindoo Bhaskara, who saw negative roots, but did not approve of them. The generalisation of the conception of quantity so as to include the negative, was an exceedingly slow and difficult process in the development of algebra.

We shall now consider the history of geometry during the Renaissance. Unlike algebra, it made hardly any progress. The greatest gain was a more intimate knowledge of Greek geometry. No essential progress was made before the time of Descartes. Regiomontanus, Xylander of Augsburg, Tartaglia, Commandinus of Urbino in Italy, Maurolycus, and others, made translations of geometrical works from the Greek. John Werner of Nürnberg published in 1522 the first work on conics which appeared in Christian Europe. Unlike the geometers of old, he studied the sections in relation with the cone, and derived their properties directly from it. This mode of studying the conics was followed by Maurolycus of Messina (1494-1575). The latter is, doubtless, the greatest geometer of the sixteenth century. From the notes of Pappus, he attempted to restore the missing fifth book of Apollonius on maxima and minima. His chief work is his masterly and original treatment of the conic sections, wherein he discusses tangents and asymptotes more fully than Apollonius had done, and applies them to various physical and astronomical problems.

The foremost geometrician of Portugal was Nonius; of France, before Vieta, was Peter Ramus, who perished in the massacre of St. Bartholomew. Vieta possessed great familiarity with ancient geometry. The new form which he gave to algebra, by representing general quantities by letters, enabled him to point out more easily how the construction of the roots of cubics depended upon the celebrated ancient problems of the duplication of the cube and the trisection of an angle. He reached the interesting conclusion that the former problem includes the solutions of all cubics in which the radical in Tartaglia's formula is real, but that the latter problem includes only those leading to the irreducible case.

The problem of the quadrature of the circle was revived in this age, and was zealously studied even by men of eminence and mathematical ability. The army of circle-squarers became most formidable during the seventeenth century. Among the first to revive this problem was the German Cardinal Nicolaus Cusanus (died 1464), who had the reputation of being a great logician. His fallacies were exposed to full view by Regiomontanus. As in this case, so in others, every quadrator of note raised up an opposing mathematician: Orontius was met by Buteo and Nonius; Joseph Scaliger by Vieta, Adrianus Romanus, and Clavius; A. Quercu by Peter Metius. Two mathematicians of Netherlands, Adrianus Romanus and Ludolph van Ceulen, occupied themselves with approximating to the ratio between the circumference and the diameter. The former carried the value $\scriptstyle {\pi }$ to 15, the latter to 35, places. The value of $\scriptstyle {\pi }$ is therefore often named "Ludolph's number." His performance was considered so extraordinary, that the numbers were cut on his tomb-stone in St. Peter's church-yard, at Leyden. Romanus was the one who propounded for solution that equation of the forty-fifth degree solved by Vieta. On receiving Vieta's solution, he at once departed for Paris, to make his acquaintance with so great a master. Vieta proposed to him the Apollonian problem, to draw a circle touching three given circles. "Adrianus Romanus solved the problem by the intersection of two hyperbolas; but this solution did not possess the rigour of the ancient geometry. Vieta caused him to see this, and then, in his turn, presented a solution which had all the rigour desirable."^[25] Romanus did much toward simplifying spherical trigonometry by reducing, by means of certain projections, the 28 cases in triangles then considered to only six.

Mention must here be made of the improvements of the Julian calendar. The yearly determination of the movable feasts had for a long time been connected with an untold amount of confusion. The rapid progress of astronomy led to the consideration of this subject, and many new calendars were proposed. Pope Gregory XIII. convoked a large number of mathematicians, astronomers, and prelates, who decided upon the adoption of the calendar proposed by the Jesuit Lilius Clavius. To rectify the errors of the Julian calendar it was agreed to write in the new calendar the 15th of October immediately after the 4th of October of the year 1582. The Gregorian calendar met with a great deal of opposition both among scientists and among Protestants. Clavius, who ranked high as a geometer, met the objections of the former most ably and effectively; the prejudices of the latter passed away with time.

The passion for the study of mystical properties of numbers descended from the ancients to the moderns. Much was written on numerical mysticism even by such eminent men as Pacioli and Stifel. The Numerorum Mysteria of Peter Bungus covered 700 quarto pages. He worked with great industry and satisfaction on 666, which is the number of the beast in Revelation (xiii. 18), the symbol of Antichrist. He reduced the name of the 'impious' Martin Luther to a form which may express this formidable number. Placing $\scriptstyle {a=1,~b=2}$, etc., $\scriptstyle {k=10,~l=20}$, etc., he finds, after misspelling the name, that $\scriptstyle {{\text{M}}_{(30)}{\text{A}}_{(1)}{\text{R}}_{(80)}{\text{T}}_{(100)}{\text{I}}_{(9)}{\text{N}}_{(40)}{\text{L}}_{(20)}{\text{V}}_{(200)}{\text{T}}_{(100)}{\text{E}}_{(5)}{\text{R}}_{(80)}{\text{A}}_{(1)}}$ constitutes the number required. These attacks on the great reformer were not unprovoked, for his friend, Michael Stifel, the most acute and original of the early mathematicians of Germany, exercised an equal ingenuity in showing that the above number referred to Pope Leo X.,—a demonstration which gave Stifel unspeakable comfort.^[23]

Astrology also was still a favourite study. It is well known that Cardan, Maurolycus, Regiomontanus, and many other eminent scientists who lived at a period even later than this, engaged in deep astrological study; but it is not so generally known that besides the occult sciences already named, men engaged in the mystic study of star-polygons and magic squares. "The pentagramma gives you pain," says Faust to Mephistopheles. It is of deep psychological interest to see scientists, like the great Kepler, demonstrate on one page a theorem on star-polygons, with strict geometric rigour, while on the next page, perhaps, he explains their use as amulets or in conjurations.^[1] Playfair, speaking of Cardan as an astrologer, calls him "a melancholy proof that there is no folly or weakness too great to be united to high intellectual attainments."^[26] Let our judgment not be too harsh. The period under consideration is too near the Middle Ages to admit of complete emancipation from mysticism even among scientists. Scholars like Kepler, Napier, Albrecht Duerer, while in the van of progress and planting one foot upon the firm ground of truly scientific inquiry, were still resting with the other foot upon the scholastic ideas of preceding ages.