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History of Mathematics After the Sixteenth Century Author(S): Raymond Clare Archibald Source: the American Mathematical Monthly, Vol History of Mathematics After the Sixteenth Century Author(s): Raymond Clare Archibald Source: The American Mathematical Monthly, Vol. 56, No. 1, Part 2: Outline of the History of Mathematics (Jan., 1949), pp. 35-56 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2304570 Accessed: 21-09-2015 13:00 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 146.186.253.206 on Mon, 21 Sep 2015 13:00:32 UTC All use subject to JSTOR Terms and Conditions HISTORY OF MATHEMATICS AFTER THE SIXTEENTH CENTURY A. THE SEVENTEENTH CENTURY The seventeenthcentury is especially outstanding in the historyof mathe- matics. It saw FERMAT lay foundationsfor modern number theory,DESCARTES formulatealgebraic geometry,PASCAL and DESARGUES open new fieldsfor pure geometry,KEPLER discover laws of heavenly bodies, GALILEO GALILEI initiate experimentalscience, HUYGENS make notable contributionsin the theory of probabilityand other fields,NEWTON create new worlds with calculus, curves, and physical observations,LEIBNIZ contributenotably in applications of the cal- culus and in mathematical notations,and NAPIER reveal a new method of com- putation. One cannot help being struck by the fact that in this century,with a single exception, all of these creators in mathematics were to be found in the north,where supremacy reigned to the end of the nineteenthcentury, in prac- tically all fields. While NAPIER, KEPLER, and GALILEO each lived many years in the sixteenth century,practically all of their importantresults to which we shall referwere obtained or announced in the seventeenth. It may be well to consider firstthe contributionsof these men. JOHNNAPIER [169] was a Scot, spent most of his life at Merchiston Castle near Edinburgh, and took an active part in the political and religious contro- versiesof the day. As BALL expressesit [1 ]: "The business of his lifewas to show that the Pope was Antichrist,but his favorite amusement was the study of mathematicsand science." In connectionwith the historyof mathematicshe is usually thought of only as the great inventor of logarithms; but the "rule of circularparts" (a mnemonicfor readily reproducing the formulaeused in solving right spherical triangles),the four formulae (known as 'NAPIER'S analogies") for solving general spherical triangles, and the calculating rods [170], called 'NAPIER'S Rods," formultiplying, dividing and taking of square roots,were also products of his genius. NAPIER seems to have had the idea of logarithms in mind as early as 1594, but it was not until 1614 that he published his researches in a "descriptionof the admirable canon of logarithms" [171]. Besides explain- ing his logarithms,he gives a table of the logarithms of natural sines from 0 to 90? foreach minute.The base of these logarithmsis qlld, whereq = (1-10-7), d=(1+' 10-7)10-7. Hence qld;(e-1(j-.j0-14) e-1. Thus it is wholly incor- rect to referto NAPERIAN logarithmsas natural logarithms [172]. While, fromthe time of ARCHIMEDES on to the time of NAPIER, there were numerousinstances of the recognitionof such a relation as am a-a= n, the law of exponents was in no sense a common idea, and it was not till later that the notion of a logarithmas an exponent became general. In NAPIER'S system the logarithmof 101 (not unity) was zero, although NAPIER recognized that a sys- tem with the logarithmof unity taken as zero was more desirable. It was ef- fected by his friend,the English mathematician,HENRY BRIGGS, whose tables 35 This content downloaded from 146.186.253.206 on Mon, 21 Sep 2015 13:00:32 UTC All use subject to JSTOR Terms and Conditions 36 OUTLINE OF THE HISTORY OF MATHEMATICS of the logarithmsof numbersfrom 1 to 20,000, and from90,000 to 100,000, to 14 places [173], were published in 1624, seven years afterNAPIER'S death. The finalpart of a great new table to 20 places of decimals has recentlyappeared in England [174] in partial (though belated) celebrationof the tercentenaryof the discovery of logarithms. In Italy NAPIER'S wonderfuldiscovery was taken up enthusiasticallyby BONAVENTURACAVALIERI, a pupil of GALILEO and professor of mathematicsat the Universityof Bologna forthe last 18 years of his life [175]; we shall presentlyhave somethingto say about his work. In Germany KEPLER'S zeal and reputationsoon broughtthem into vogue there. Let us now consider some of the contributionsmade by the second man of our group JOHANNKEPLER [176], mathematician,mystic, one of the founders of modern astronomyand an exceptional genius, outstandingas a brilliantcal- culator and patient investigator.In 1601 at Prague he became the assistant of the remarkable Danish astronomical observer TYGE BRAHE [176], who died a little later. The vast body of accurate observationsof the planets which Kepler thus inheritedled him in 1609 and 1619 to formulatethe followinglaws: 1. All the planets move round the sun in elliptic orbits with the sun at one focus; II. The line joining the planet to the sun sweeps out equal areas in equal intervals of time; III. For all planets the square of the time of one complete revolution is proportionalto the cube of the mean distance fromthe sun. Purely as a thrillingintellectual experience,without any imaginable prac- tical application, APOLLONIUS of Perga, and otherGreeks, developed a marvell- ous body of knowledgewith regardto conic sections. Then suddenly, 1800 years later, to a KEPLER this knowledgehad most illuminatingpractical applications. KEPLER'S contributionsto the infinitesimalcalculus included findingvol- umes of surfacesof revolutionsof conics about lines (93 differentsolids including the torus); deriving,in effect,the value of f sin tdt,and other integrals; and in solving maximum-minimumproblems concerning cylinders, cones, and wine barrels.Of the fourknown star polyhedraKEPLER discoveredtwo, the othertwo being found centurieslater (1809) by Louis POINSOT.Star polygons and prob- lems of fillinga plane and space by regularfigures were studied by KEPLER. He discovered how to determinewhether a conic is a hyperbola,ellipse, parabola, or circle when given a vertex, the axis through it, and an arbitrarytangent with its point of contact. In one of his works, which consists rather in the enunciation of certain general principles,illustrated by a few cases, than in a systematicexposition, he laid down what has been called a Principle of Conti- nuity,and gives as an example the statementthat a parabola is at once the limit- ing case of an ellipse and of a hyperbola. He illustratedthe same doctrine by referenceto fociof conics; he explained also that parallel lines should be regarded as meetingat infinity.KEPLER contributedto the simplificationof computations and published a volume of logarithms(1624-1625). The last memberof our group, GALILEO GALILEI [1771,one of the most inter- This content downloaded from 146.186.253.206 on Mon, 21 Sep 2015 13:00:32 UTC All use subject to JSTOR Terms and Conditions HISTORY OF MATHEMATICS AFTER THE SIXTEENTH CENTURY 37 esting figuresin the historyof science, was the founderof the science of dy- namics. His work in astronomywe shall not consider. A native of Pisa, he is said to have carried on experimentsfrom the leaning tower there and to have arrived at the law that the distance of descent of a falling body was propor- tional to the square of the time, in accordance with the well-knownlaw that S = igt2. GALILEOwas the firstto realize that, neglectingair resistance,the path of a projectile is a parabola. He speculated interestinglyon laws involvingmo- mentum. He suggested that arches of bridges should be built in the form of cycloids [178], and by weighingpieces of paper he is believed to have found that the area of a cycloid is less than three times that of its generatingcircle. From 1609 on, he constructedtelescopes. Anothermathematician who lived many years in the sixteenthcentury, al- though his outstanding publication appeared in the seventeenthcentury, was the Englishman THOMAS HARRIOT [179], who is of special interest to Amer- icans since he was sent in 1585 to survey and map Virginia,now North Caro- lina. He died in 1621; one of his works published in 1631, Artis Analyticae Praxis, deals with algebraic equations of the first,second, third, and fourth degrees. It is more analytic than any algebra that preceded it, and marks an advance both in symbolismand notation, though negative and imaginaryroots are rejected. It was potentin bringinganalytic methodsinto generaluse. Harriot was the firstto use the signs > and < to representis greater than and is less than. When he denoted the unknown quantity by a, he representeda2 by aa, a3 by aaa, and so on. This is a distinct improvementon Vieta's notation. In unpublished manuscripts,attributed to HARRIOT, there is a table of binomial coefficientsworked out in the form of a PASCAL triangle [180]. It was more than thirtyyears afterHARRIOT'S death that PASCAL is known to have used this triangle. MORLEY stated that the manuscriptscontained a well-formedanalyt- ical geometry,but D. E. SMITH found that the manuscriptin question was not in HARRIOT's handwriting. HARRIOT'Sposthumous work appeared in the same year, 1631, as the first edition of the Clavis Mathematicaeof WILLIAM OUGHTRED [181], a work on arithmeticand algebra. After NAPIER'S Descriptio in 1614, this was the most influentialmathematical publication in Great Britain, in the firsthalf of the seventeenth century.
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