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Vincent Riccati, S.J.
(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 (ex + e-x)/2 but their inventor, the Jesuit, Vincent Riccati (son of Jacob) developed them and proved their consistency using only the geometry of the unit hyperbola x2 - y2 = 1 or 2xy = 1. 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 e and the function ex ten years previous to Riccati's book.

Figure 1 The trigonometric functions
are taken from the unit circle
Figure 2 The hyperbolic functions
are taken from the unit hyperbola
Figure 3: equivalent areas: AOK = FOH Figure 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 hyperbolas xy = k and evident in the figure. All right triangles OAK have the same area k/2 no matter what point A is chosen on the curve. This property enables us to replace area measure by linear measure OK, 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.

Given: G/K = P/H, then
AKGE = S dx/2x
= .5 log(G/K)
= .5 log (P/H)
= S dx/(2x)
Adding and subtracting then just involve constructing the correct proportions, i.e., finding the proper OP and OG for the proportions:
OK x OP = OG x OH which represents
adding: AKPN = AKHF + AKGE
OG/OK = OP/OH which 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


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 of Guldin'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

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Jesuit history, tradition and spirituality

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