6 Dec 2008

Barrow's Linelets, Timelets and Differential Triangle

by Corry Shores
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Isaac Barrow (1630 - 1677), who was interested in the notion of motion at a point, conceived of time as a "mathematical quantity measurable by, although not dependent on, motion, and upon the suggestion of sensory evidence he thought of it as a continuous magnitude, 'passing with a steady flow" (Boyer 180b).

He did not use the limit concept when dealing with the problems regarding continua and instantaneous velocity, instead blending atomistic and kinematic perspectives. He says in his Geometrical Lectures:

To every instant of time, or indefinitely small particle of time, (I say instant or indefinite particle, for it makes no difference whether we suppose a line to be composed of points or of indefinitely small linelets; and so in the same manner, whether we suppose time to be made up of instants or indefinitely minute timelets); to every instant of time, I say, there corresponds some degree of velocity, which the moving body is considered to possess at the instant. (180c)

Barrow considered time and the line to be analogous, and he thought that these magnitudes can be considered as being either an aggregate of instants or points, or as the continuous flow of one instant or point. But like Cavalieri, Barrow preferred to conceive time as composed of indivisibles. In fact, Barrow defended Cavalieri's Method of Indivisibles (181a.b).

Regardless of his views on continua, he devised a technique much like the modern method of differentiation. Consider his Differential Triangle:

Justify Full
Barrow's method for finding the tangent to the curve is much like the one used today, with Δy and Δx in place of his a and e. As well, in the rules for his calculation, he says that we "omit all terms containing a power of a or e, or products of these (for these terms have no value)," and we reject "all terms consisting of letters denoting known or determined quantities, or terms which do not contain a or e" (Boyer 183a, image Eves 395d).

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.

Eves, Howard. An Introduction to the History of Mathematics. London: Brooks/Cole - Thomson Learning, 1990.

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