By Edoardo Ballico, Ciro Ciliberto

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XII, App. "integration by parts" I have noticed . G. |^/ N/(^ + 2 = M{^ + v/(^ + ) 2 5 2 \cos I, = ^l g{(i+sme}l(i-sme}} 2 )}/] I, 8 9 32 BARROW'S GEOMETRICAL LECTURES Graphical Integration of any Function For any function, f(x), that cannot be integrated by the foregoing rules, Barrow gives a graphical method for ^f(x}dx as a logarithm of the quotient of two radii vectores of the curve r=f(6), and for their reciprocals He He . . \dxjf(x} as a difference of Lect. XII, App. Ill, 5-8 Fundamental Theorem in Rectification Lect.

The similarity of the two methods of Barrow and Newton is far too close to admit of them being anything else but the outcome of one single idea and I argue from the dates given above that Barrow had developed most of his geometry from the researches begun for the necessities of lectures at know that Barrow's work on the Gresham College. difficult theorems and problems of Archimedes was largely a suggestion of a kind of analysis by which they were reduced ; We What is then more to their simple component problems.

LECTURE along any other is is 43 keeping parallel to line, and called the genetrix, which II said to be applied to the latter is called the directrix the former itself; by these are described ; paral- lelogrammatic surfaces, when the genetrix and the directrix are both in the same sary, plane, and prismatic and cylindrical In general, the genetrix may, surfaces otherwise. 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.

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