Search Loci: Convergence:
[T]here is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being.
Presidential Address to British Association, 1869.
Thomas Simpson and Maxima and Minima
Maximizing y as an implicit function of x
To find the greatest value of y in the equation a4 x2 = (x2 + y2 )3 (Example XX, p. 42).
Assuming that y is a function of x, implicit differentiation ( Simpson says: "by putting the whole equation into fluxions") leads to 2a4 x = 3(x2 + y2 )2 (2x + 2yy¢). But we have to make y¢ = 0, so 2a4 x = = 3(x2 + y2 )2 (2x). Thus a2/Ö3 = x2 + y2, which in turn becomes a6/3Ö3 = (x2 + y2)3. Since (x2 + y2 ) = a4 x2, we can conclude that a6/3Ö3 = a4 x2, whence
Replacing this value in the original expression we get
An alternative approach, also presented in Doctrine, goes as follows: Taking the cube root we arrive at a4/3 x2/3 = x2 + y2, thus y2 = a4/3 x2/3 - x2. Consequently, 2yy¢ = (2/3)a4/3x-1/3 - 2x, i.e.
Remark: An aspect not considered by Simpson is whether a maximum is actually attained by y at
First of all, let us note that he is undoubtedly working with the "positive part" of y defined by
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