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Noted that the index notation is rarely used, at least not till very late in the book in places where he could do nothing else, although Wallis had used even fractional indices a dozen In a later lecture we have the truly terrifying years before. equation (rrkk rrff-\- 2fmpa)lkk Again we must note the = (rrmm + ^fn^pa)|kk. fact that all Barrow's work, BARROW'S GEOMETRICAL LECTURES 12 without exception, was geometry, although it is fairly evident that he used algebra for his own purposes. From the above, it is quite easy to see a reason why Barrow should not have turned his work to greater account ; but in estimating his genius one must make allowance for this disability in, or dislike for, algebraic geometry, read work what could have been got out of it (what I Newton and Leibniz got out of it), and It not stop short at just what was actually published.

E. F. \ Q a*dx = k(a x - Lect. i) ) Ptan0d0=bg(cos0) ^ecOde . XII, App. 3, Prob. 3, 4 Lect. XII, App. - x2 = ) {log (a + x)/(a-x)}/2a \dx/(a = d(tan 0} dO \tan d(cos 6) (see Form D) dd - tan 6 cos 0, both being equal to ^secOdO, the only example of Lect. 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 .

Results, The remaining standard forms Barrow is unable apparently to obtain directly; and the same remark applies to the rest of the laws. So he proceeds to show that Differentiation and Integration are inverse operations. z = [ydx, then R. dz/dx =y . . (L^, Lect. dz/dx=y, then fydx Hence the standard forms for integration are to be obtained immediately from those already found for differentiation. Barrow, however, proves the integration formula for an integral power independently, in the course of certain theorems in Lect.