146 A History ofMathematics
the most significant event was Stevin’s propagandist work (La Dismeof 1585). Here thereisa
possible debt to the Islamic world, specifically to al-K ̄ash ̄i, but we are in need of further evidence.
However, even given the various possible lines of transmission from Islamic mathematics, an
analysis of what happened in the sixteenth century must take into account not just its ‘influences’,
but its own particular momentum and early-modern ideology; Stevin was an early enthusiast for
decimalization, who hoped to replace both ‘astronomers’ numbers’ (sexagesimals) and the confused
systems of measurement with which surveyors were faced.
And the surveyor or land-meter...is not ignorant (specially whose business and employment is great) of the trouble-
some multiplication of rods, feet, and oftentimes of inches, the one by the other, which not only molests, but also
often...causes error, tending to the damage of both parties...(Stevin 1958, p. 395)He also was responsible for producing tables of compound interest, in which again decimals
simplified the task tremendously.
The new algebra, if we accept Klein’s thesis, has a generally accepted ‘starting point’: the redraft
by Bombelli of his algebra textbook of 1560 (published 1572). Having been shown a manuscript of
Diophantus, Bombelli changed his emphasis to accord better with his ancient model, removing the
traditional practical problems and replacing them by ones taken from Diophantus. This ‘moment’—
a change in the idea of number which overthrows many of the ancient Greek ideas in the interest
of what is simple and practical—we could call a first mathematical revolution (to answer the
first of the questions which we posed in section 1); the second is the gradual, equally un-Greek
introduction of infinitesimal processes.Exercise 4.(a) Use Tartaglia’s method to solve the equation ‘cube and three things equal to four’, or
x^3 + 3 x= 4. (Hint:You are given u−v and uv; find u+v.) (b) Why do you not get the obvious answer
1? (c) Try to prove that x as given in Tartaglia’s formulation is a solution of the general cubic equation
‘cube and things equal to numbers’ (1) by algebra and (2)—if you have the patience for it—by geometry,
as Cardano did.Exercise 5. Solve ‘cube and thing equal 500’ (as in the question of the sapphire and the ducats) by
Tartaglia’s recipe.7. On authority
Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbarians, that I considered
it necessary, in order to introduce an entirely new form into it, to think it out and publish a new vocabulary, having
got rid of all its pseudo-technical terms lest it should retain its filth and continue to stink in the old way...And yet
underneath the Algebra or Almucabala which they lauded and called ‘the great art’, all Mathematicians recognized
that incomparable gold lay hidden, though they used to find very little. (Viète,The Analytic Art, in Klein 1968,
pp. 318–9)
It has become a matter of common usage to call the barbarous age that time which extends from about 900 or a
thousand years up to about 150 years past, since men were for 700 or 800 years in the condition of imbeciles without
the practice of letters or sciences...but although the afore-mentionedprecedingtimes could call themselves a wise
age in respect to the barbarous age just mentioned, nevertheless we have not consented to the definition of such a
wise age,since both taken together are nothing but the true barbarous agein comparison to that unknown time at which
we state that it [that is, the true wise age] was, without any doubt, in existence. (Stevin,Géographie, quoted in Klein
1968, p. 187)