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:

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.

Cavalieri's Indivisibles

by Corry Shores
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Cavalieri concieved of a surface as made up of an indefinite number of equidistant parallel lines and of a solid as composed of parallel equidistant planes, these elements being designated the indivisibles of the surface and of the volume respectively.

(Boyer 117c).

Cavalieri says the number of these indivisbles must be "indefinitely great," but he remained "agnostic" as to the nature of infinity (Hegel distinguishes Cavalieri's notion of indivisibles with the vanaishing divisibles of Newton).

Cavalieri says that if one were to rotate the set of indivisibles (parallel lines) making up a planar shape, then the area of the new planar piece will equal the original. Likewise if we slid the parallel planar sections of a solid, we would obtain another solid with the same area:

This last result can be strikingly illustrated by taking a vertical stack of cards and then pushing the sides of the stack into curved surfaces; the volume of the disarranged stack is the same as that of the original stack.

(Eves 387d).

The reason, then, that Cavalieri did not relate these indivisibles to the notion of infinity was because he focused more on the correspondaence between the indivisibles of the two configurations, rather than upon the totality of indivisibles within a single area or volume.

But still the applications of Cavalieri's indivisibles imply some notion of infinity, as for example when demonstrating that a parallelogram is made up of the sum of the double of the lines of either of its internal triangles:

The parallelogram is divided into two triangles by a diagonal line. Then, one forms the smaller triangles in the corners by marking off equal lines, BC and EF, then drawing lines BM and and HE parallel to CD. These lines are equal because their internal angles must be equal. Thus for every parallel line in the one triangle, there is an equal counterpart in the other, hence the double the sum of all the indivisibles in one triangle equals the sum of all lines in the whole parallelogram (Boyer 118b-119).

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

Eves, Jamie. An Introduction to the History of Mathematics, with Cultural Connections. London: Brooks/Cole - Thomson Learning, 1990.