"In 1669 [Isaac Barrow] issued his Lectiones opticæ et geometricæ: this, which is his only important work, was republished with a few minor alterations in 1674. A complete edition of all Barrow's lectures was edited for Trinity College by W. Whewell, Cambridge, 1860. It is said in the preface to the Lectiones opticæ et geometricæ that Newton revised and corrected these lectures adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. ... The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves. Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it [the tangent line] was known; hence if the length of the MT could be found (thus determining the point T) then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn he got a small triangle PQR (which he called the differential triangle because, its sides PR and PQ were the differences of the abscissas and ordinates of P and Q) so thatTM : MP = QR : RP.To find QR : RP he supposed that x,y were the coordinates of P and x - e, y - a those of Q. ...Using the equation of the curve and neglecting the squares and higher powers of e and a as compared with their first powers he obtained e : a The ratio a/e was subsequently (in accordance with a suggestion made by de Sluze) termed the angular coefficient of the tangent at the point. Barrow applied this method to the following curves (i) x^2 (x^2 + y^2) = r^2y^2; (ii) x^3 + y^3 = r^3; (iii) x^3 + y^3 = rxy, called la galande; (iv) y = (r - x) tan\frac{\pi x}{2r}, the quadratrix; and (v) y = r \tan \frac{\pi\,x}{2r}. ...take as an illustration the simpler case of the parabola y^2 = px. Using the notation given above we have for the point P, y^2 = px; and for the point Q, (y - a)^2 = p(x - e). Subtracting we get 2ay - a^2 = pe. But if a is an infinitesimal quantity, a^2 must be infinitely smaller and may therefore be neglected: hence e : a = 2y : p. Therefore TM : y = e : a = 2y : p. That is TM = \frac{2y^2}{p} = 2x. This is exactly the procedure of the differential calculus, except that we there have a rule by which we can get the ratio \frac{a}{e} or dy \over dx directly without the labour of going through a calculation similar to the above for every separate case."