By Bibhutibhushan Datta
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VIII, 1 5 Ify is any function of x, and z = a + fry, Lect. IX, i then dzjdx = b dyfdx n = n~ r r is function and z a of x, 7. y Lect. (i/y)- dyjdx n 1 8. Special case d(x }jdx = n x"' or n (y/x), Lect. IX, 4 where is a positive rational 9. The case when n is negative is to be deduced from the combination of Forms i and 8. 2 10. If y = tanx, then dy/dx = sec x, proved as Ex. 5 on " " differential at the end of Lect. X. the triangle n. It is to be noted that the same two figures, as used for tan x, can be used to obtain the differential coefficients or dzjdx .
I say that this is possible, but I do not think it is at all probable; for it is to be noted that Barrow's description of the method is in the first person singular (although, when giving the reason for its introduction, he says "frequently used by us"); and remembering the authentic accounts of Barrow's conscientious honesty, and also judging by the later work of Newton, I think that the only alternative to be considered is that first given. Also, if that is accepted, we" have a natural explanation of the lack of what I call the Barrow could see the true appreciation of Barrow's genius.
Be taken as a curve, which is if neces- intended to include polygons, and the genetrix and the directrix may usually The same kinds of assumptions are be interchanged. made for simple motions of rotation; and by these are described circles and rings and sectors or parts of these, when a straight line rotates in in itself or in the line (in the own plane about its produced if ; the directrix wider sense given above), and does not plane of the genetrix, of which one point be is is a point a curve lie in the supposed to fixed, the surfaces generated are pyramidal or conical.