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(1707 - 1775)

and his hyperbolic functions

Vincent Riccati, S.J. was born in Castel-Franco, Italy. He worked together with Girolamo Saladini in publishing his discovery, the hyperbolic functions - although Lambert is often incorrectly given this credit. Riccati not only introduced these new functions, but also derived the integral formulas connected with them, and then, still using geometrical methods. He then went on to derive the integral formulas for the trigonometric functions. His book Institutiones is recognized as the first extensive treatise on integral calculus. The works of Euler and Lambert came later.

Saladini and Riccati also considered other geometrical problems, including the tractrix, the strophoid and the four-leaf rose introduced by Guido Grandi. His father Jacobi (after whom is named the Riccati differential equation) was one of the principal Italian mathematicians of the century, and his brother Giovanni was also a prominent mathematician.

Today we treat the hyperbolic functions as pairs of exponential functions(ebut their inventor, the Jesuit, Vincent Riccati (son of Jacob) developed them and proved their consistency using only the geometry of the unit hyperbola^{x}+ e^{-x})/2x. Vincent Riccati followed his father's interests in differential equations which arose naturally from geometrical problems. This led him to a study of the rectification of the conics in Cartesian coordinates and to an interest in the areas under the unit hyperbola. Riccati developed the properties of the hyperbolic functions from purely geometrical considerations. He used geometrical motivation even though he was familiar with the work of Euler, who had introduced the symbol and concept of the natural number^{2}- y^{2}= 1 or 2xy = 1eand the functioneten years previous to Riccati's book.^{x}

Figure 1 The trigonometricfunctions

are taken from the unit circleFigure 2 The hyperbolicfunctions

are taken from the unit hyperbolaFigure 3: equivalent areas: AOK = FOHFigure 4: AKGE = AKPN - AKHF

Riccati's development of the hyperbolic functions may be described as follows. The algebra concerning circular sectors is simple since the arc length is proportional to the area of the sector. Hyperbolas, however, do not have this property, so a different algebra is needed. The latter arises naturally from a property peculiar to hyperbolasxy = kand evident in the figure. All right trianglesOAKhave the same areak/2no matter what pointAis chosen on the curve. This property enables us to replace area measure by linear measureOK, thus introducing a kind of logarithm.

It can be seen from figures 3 and 4 that the following areas are equivalent: AOK = FOH because of the property just mentioned, AOQ = QKHF by subtracting the common area OKQ, AKHF = AOF by adding AQF to each. So for any point F on the curve, the area of the sector AOF equals the area under AF. This area function depends on point F as well as on H, the projection of F on the x-axis. Addition and subtraction of areas can then be accomplished by multiplication and division of distances along the x-axis, and vice versa.Adding and subtracting then just involve constructing the correct proportions, i.e., finding the proper

Given:G/K = P/H, then

AKGE= S dx/2x

= .5 log(G/K)

= .5 log (P/H)

= S dx/(2x)

=FHPNOPandOGfor the proportions:

OK x OP = OG x OHwhich represents

adding:AKPN = AKHF + AKGE

OG/OK = OP/OHwhich represents

subtracting:AKGE = AKPN - AKHF

From these definitions of the hyperbolic functions and the logarithm operations, it is possible to derive a complete array of formulas regarding the double and triple arguments and from them to deduce other formulas for the square and cube roots. Riccati goes even further and proves the integral formulas for trigonometric as well as hyperbolic functions.

Some other innovative works of Riccati include his treatment of the location of points that divide the tangents of a tractrix and also the problem posed by Guido Grandi about the four-leaf rose.## Some of the figures used in Riccati's book

References

Archivum Historicum Societatis Iesu ( AHSI ) Rome: Institutum Historicum

Bangert, William A History of the Society of Jesus. St. Louis: St. Louis Institute, 1972uis, 1810

Boyer, Carl A history of mathematics. New York: Wiley, 1968 Florian Cajori: A History of Mathematical Notation. Chicago: Open Court,1928, p.172.

Gillispie, Charles. C. ed., Dictionary of Scientific biography. 16 vols. New York: Charles Scribner and Sons, 1970

{ Reference to V. Riccati in the Dictionary of Scientific Biography is found in v 11 p401-402. Siegmund Gunther: Die Lehre von den gewohnlichen und verallgemeinerten Hyperbelfunktionem. Halle: Nebert, 1888, p. 1 - 55. MacDonnell, S.J Joseph.: "Using sinh and cosh to solve the cubic" in the International Journal of Mathematical Education. London: vol 18 #1 1987, p. 111-118. Moritz Cantor: Vorlesungen uber Geschichte der Mathematik. vol. 1, Leipsig: 1892, Teubner, p. 411. Charles Naux: Histoire des logarithmes. Paris: 1871, p. 124- 149

Oldenburg, Henry ed. Philosophical Transactions of the Royal Society. vols. 1-30. London: 1665-1715

Reilly, Conor "A catalogue of Jesuitica in the Philosophical Transactions of the Royal Society of London" in A.H.S.I. vol. 27,1958, p. 339-362

Sarton, GeorgeThe study of the history of mathematics. Cambridge, Mass: Harvard, 1936

Sommervogel, Carolus Bibliothèque de la compagnie de Jésus. 12 volumes. Bruxelles: Société Belge de Libraire, 1890-1960

{28 entries are found in Sommervogel; some examples are the following:

{Tomus primus opuscuorum (Bologna, 1757)

{Institutiones analyticae (Bologna, 1765)}

## Adventures of Some Early Jesuit Scientists

José de Acosta, S.J. - 1600: Pioneer of the Geophysical Sciences

François De Aguilon, S.J. - 1617: and his Six books on Optics

Roger Joseph Boscovich, S.J. - 1787: and his atomic theory

Christopher Clavius, S.J. - 1612: and his Gregorian Calendar

Honoré Fabri, S.J. - 1688: and his post-calculus geometry

Francesco M. Grimaldi, S.J. - 1663: and his diffraction of light

Paul Guldin, S.J. - 1643: applications ofGuldin's Rule

Maximilian Hell, S.J. - 1792: and his Mesmerizing encounters

Athanasius Kircher, S.J. - 1680: The Master of a Hundred Arts

Francesco Lana-Terzi, S.J. - 1687: The Father of Aeronautics

Francis Line, S.J. - 1654: the hunted and elusive clock maker

Juan Molina, S.J. - 1829: The First Scientist of Chile

Jerôme Nadal, S.J. -1580: perspective art and composition of place

Ignace Pardies, S.J. - 1673: and his influence on Newton

Andrea Pozzo, S.J. - 1709: and his perspective geometry

Vincent Riccati, S.J. - 1775: and his hyperbolic functions

Matteo Ricci, S.J. - 1610: who brought scientific innovations to China

John Baptist Riccioli, S.J. - 167I: and his long-lived selenograph

Girolamo Saccheri, S.J. - 1733: and his solution to Euclid's blemish

Theorems of Saccheri, S.J. - 1733: and his non Euclidean Geometry

Christopher Scheiner, S.J. - 1650: sunspots and his equatorial mount

Gaspar Schott, S.J. - 1666: and the experiment at Magdeburg

Angelo Secchi, S.J. - 1878: the Father of Astrophysics

Joseph Stepling, S.J. - 1650: symbolic logic and his research academy

André Tacquet, S.J. - 1660: and his treatment of infinitesimals

Pierre Teilhard de Chardin, S. J. - 1955: and The Phenomenon of man

Ferdinand Verbiest, S.J. - 1688: an influential Jesuit scientist in China

Juan Bautista Villalpando, S.J. - 1608: and his version of Solomon's Temple

Gregory Saint Vincent, S.J. - 1667: and his polar coordinates

Nicolas Zucchi, S.J. - 1670: the renowned telescope maker## Influence of Some Early Jesuit Scientists

The 35 lunar craters named to honor Jesuit Scientists: their location and description

Post-Pombal Portugal opinion of Pre-Pombal Jesuit Scientists: a recent conference

Seismology, The Jesuit Science. a Jesuit history of geophysics

## Another menu of Jesuit Interest

Jesuit history, tradition and spirituality

Visit the Jesuit Resource Page for even more links to things.Jesuit

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