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72 THE NATURE OF MATHEMATICS 

their fluxions, either alone or combined with the co-ordiuates 
themselves. This is equivalent to Leibniz's general problem 
of integration, and is the problem to which we saw, at the end 
of the fourth chapter, that theoretical astronomy reduces. 

Newton denoted the fluxion of x by "a;," and the fluxion of 
the fluxion (the acceleration) of x by "a." It is obvious that 
this notation becomes awkward when we have to consider 
fluxions of higher orders ; and further, Newton did not indicate 
by his notation the independent variable considered. Thus 

"y" might possibly mean either -^ or 3?. We have x = , 

dt ax dt 

fjsv* d jf* fJ^y 

x = = T- ; but a dot-notation for would be clumsy and 
dt dt- ' dt n 

inconvenient. Newton's notation for the "inverse method 
of fluxions " was far clumsier, even, and far inferior to 
Leibniz's "/". 

The relations between Newton and Leibniz were at first 
friendly, and each communicated his discoveries to the other 
with a certain frankness. Later, a long and acrimonious dis- 
pute took place between Newton and Leibniz and their 
respective partisans. Each accused unjustly, it seems the 
other of plagiarism, and mean suspicious gave rise to meanness 
of conduct, and this conduct was also helped by what is some- 
times called " patriotism." Thus, for considerably more than 
a century, British mathematicians failed to perceive the great 
superiority of Leibniz's notation. And thus, while the Swiss 
mathematicians, James Bernoulli (1654-1705), John Bernoulli 
(1667-1748), and Leonhard Euler (1707-1783), the French 
mathematicians d'Alembert (1707-1783), Clairaut (1713- 
1765), Lagrange (1736-1813), Laplace (1749-1827), Legeudre 
(1752-1833), Fourier (1768-1830), and Poisson (1781-1850), 
and many other continental mathematicians were rapidly l 

1 It is difficult for a mathematician not to think that the sudden 
and brilliant dawn on eighteenth century France of the magnificent 
and apparently all-embracing physics of Newton and the wonder- 
fully powerful mathematical method of Leibniz inspired scientific 
men with the belief that the goal of all knowledge was nearly 
reached and a new era of knowledge quickly striding towards 
perfection begun ; and that this optimism had indirectly much to 
do in preparing for the French Revolution.