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# Issac Barrow (1630-1677)

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## Denise Ramirez

on 14 October 2014

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#### Transcript of Issac Barrow (1630-1677)

Issac Barrow (1630-1677)
In 1662 he was made professor of geometry at Gresham College. For the remainder of his life he devoted himself to the study of divinity. He was appointed master of Trinity College in 1672, and held the post until his death.
Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known; hence, if the length of the subtangent 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 abscissae and ordinates of P and Q), so that K
TM : MP = QR : RP.
To find QR : RP he supposed that x, y were the co-ordinates of P, and x − e, y − a those of Q (Barrow actually used p for x and m for y, but this article uses the standard modern notation). Substituting the co-ordinates of Q in 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 Sluze) termed the angular coefficient of the tangent at the point.

Barrow applied this method to the curves

x2 (x2 + y2) = r2y2, the kappa curve;
x3 + y3 = r3;
x3 + y3 = rxy, called la galande;
y = (r − x) tan πx/2r, the quadratrix; and
y = r tan πx/2r.
It will be sufficient here to take as an illustration the simpler case of the parabola y2 = px. Using the notation given above, we have for the point P, y2 = px; and for the point Q:

(y − a)2 = p(x − e).
Subtracting we get

2ay − a2 = pe.
But, if a be an infinitesimal quantity, a2 must be infinitely smaller and therefore may be neglected when compared with the quantities 2ay and pe. Hence

2ay = pe, that is, e : a = 2y : p.
Therefore

TM : y = e : a = 2y : p.
Hence

TM = 2y2/p = 2x.
This is exactly the procedure of the differential calculus, except that there we have a rule by which we can get the ratio a/e or dy/dx directly without the labour of going through a calculation similar to the above for every separate case.
Isaac Barrow was born in London in 1630 He went to school first at Charterhouse , and subsequently to Felstead. He completed his education at Trinity College, Cambridge; after taking his degree in 1648,Barrow studied arithmetic, geometry and optics.

Issac Barrow Died: May 4, 1677, In London, United Kingdom
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