Introduction
After outlining the state of algebra and geometry at the beginning
of the sixteenth century, we move to discuss the advances in these fields
between 1500 and 1640. A separate section is devoted to the development and
use of algebraic geometry by Descartes, Fermat and Newton. We close with an
attempt to assess the relative importance of these developments.
State of the Arts: Chuquet and
Pacioli
At the beginning of the sixteenth century,
mathematics was dominated by its Greek heritage and therefore by the study
of geometry. But algebra was not wholly absent, and significant advances in
notation had been made towards the end of the fifteenth century. Two works
were particularly important in this regard: Nicolas Chuquet’s
(c.1440-c.1488) Triparty (1484) and Luca Pacioli’s (c.1445-1517) Summa
(1494). Pacioli’s symbolism was limited, consisting mostly of
abbreviations. Although Chuquet’s symbolism was more advanced, the
influence of this work was limited by its very small circulation: it was
not properly published until 1880.
Algebraic Advances: Cardano,
Bombelli, Viète and Harriot
The crucial advances on cubic equations were made
in the secretive climate of sixteenth century Italian academia. The story
of their solution is complicated and involves many interesting biographical
details.[1]
Let it here suffice to say that Girolamo Cardano (1501-76) and his student
Ludovico Ferrari (1522-65) were together able to solve not only all
thirteen types of cubic equation, but all twenty types of quartic equation.
The results were published in Cardano’s Ars Magna (1545).
Remarkably, these advances were made using the old rhetorical algebra, in
which, for example, x3 + cx = d was written as cube
and things equal to numbers. The resulting proofs were very long.
Cardano accepted the negative
solutions to which his rules sometimes led, though he described them as
“fictitious”. He struggled, however, when his rules required him to find the
square root of a negative number, stating that such numbers as √-15
are as “refined as they are useless”. Some of these difficulties were later
ameliorated by Rafael Bombelli (1526-72) whose Algebra (1572)
included the first discussion of what we now call ‘complex numbers’.[2] Bombelli also developed an algebraic notation
similar to that of Chuquet.
Although his algebra was syncopated
rather than symbolic, further notational advances were made by François
Viète (1540-1603). His Isagoge (1591) used letters not only for
unknowns but also for general coefficients.
Another who did important work on
cubics and quartics was Thomas Harriot (1560-1621). Expressed in modern
notation, Harriot was first to see that the three solutions of (x – a)(x
– b)(x – c) = 0 are x = a, x = b and x = c.[3] He also made
advances in notation.
Progress in Geometry: Harriot,
Roberval, Desargues and Pascal
Harriot also achieved the rectification and quadrature
of the equiangular spiral and in about 1630, Gilles Personne de Roberval
(1602-75) showed that the area under one arch of the cycloid is three times
that of the generating circle.
Probably the most significant
development in geometry over the period in question was that of Girard
Desargues’ (1591-1661) projective geometry. Projective geometry is
concerned with geometrical properties that are invariant under projection.
Desargues’ insight, presented in his Brouillon Project (1639), was
that viewed from the vertex of a cone, all sections of that cone appear the
same; they all appear as a circle. All conics are therefore projections of
the circle. Among other things, this means that many theorems about circles
can be interpreted as theorems about other conics, and visa-versa.
Blaise Pascal (1623-62) used
Desargues’ methods to prove what is now known as Pascal’s theorem. The
theorem states that if a hexagon is inscribed within a conic, then the
‘meets’ of lines produced from opposite sides of the hexagon lie on a
straight line.
Algebraic Geometry: Descartes,
Fermat and Newton
René Descartes’’ (1596-1650) La Géométrie,
one of three appendices to his Discours de la Méthode (1637), introduced the world to algebraic
geometry. In effect, the method was to set up a system of axes, chosen
according to the problem in hand, relative to which the position of any
point in the plane may be specified by two coordinates, x and y,
thus allowing the problem to be investigated algebraically.
The power of Descartes’ method is illustrated by his
treatment of Pappus’ problem. That problem involves finding the locus of
points P such that, given four lines in a plane, the product of P’s
distances from two of the lines stands in a specified ratio to the product
of P’s distances from the other two lines, where those distances are
measured at given angles to the lines.
Pappus had stated that the locus of P was a conic section, a result
that Descartes was able to prove. Algebraic analysis of Pappus’ problem
leads to expressions for the distances between P and the given lines which,
substituted into equation (1) above, yield a second degree equation
specifying the locus of P in terms of the relationship between x and
y. That equation allows us to find values of y corresponding
to any given value of x, and thus to plot the curve.
Descartes also applied the method to the equivalent
problem involving five, six or more lines. In these cases the equation
which represents the locus could have a degree of greater than two.[4]
A similar system was developed
independently by Pierre de Fermat (1601-65). Fermat stated that equations
in two unknowns represent curves in the plane, and used a system of
rectangular axes. Although the work was not published in his lifetime, many
of the ideas were formulated before the appearance of Descartes’ La
Géométrie.[5]
Indeed, while this branch of mathematics (Cartesian geometry) bears
Descartes’ name, “Fermat’s approach is much closer to the modern treatment
of the subject”.[6]
Isaac Newton (1642-1727) used
Descartes’ system in his study of cubics in the 1660’s. Newton showed that
as many as 72 different types of curve could be represented by cubic equations
(later mathematicians added another six). Although seemingly unaware of
Desargues’ work, Newton went on to claim that every cubic could be seen as
the shadow (i.e. projection) of one of just five different cubics.
The Significance of these
Developments
While Desargues’ projective geometry was put to
good use by Pascal and others, these methods were not widely used until
their rediscovery in the nineteenth century.[7] And if the sixteenth century
began with the dominance of geometry, then by the end of the seventeenth
century algebra was in the ascendancy.
Viète and his contemporaries had
regarded algebra as a method of analysis which should be supplemented by
synthesis using the classical geometric methods.[8] Even the formulas for solving
cubic and quartic equations were originally proved by geometric means.[9]
However, the study of these
equations not only showed an increased proficiency in algebra, it also marked
a move away from geometric thinking; unlike x, x2
and x3, x4 has no straightforward
geometric interpretation. Moreover, it was these researches that first
forced mathematicians to think about complex numbers.
While algebra was already making inroads into the
dominance of geometry, it was the power of Descartes’ algebraic methods,
especially applied to higher plane curves (those other than straight lines
and conics), that led mathematicians to both use and trust algebra.
Mathematicians came to think that algebraic analysis need not be
supplemented by geometric synthesis. The methods had not only created new
ties between algebra and geometry, they had encouraged mathematicians to
grant algebra an authority of its own.
Notes
Last Updated 23rd
January 2004
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