The History of Mathematics: A Brief Course

1. PRELUDE TO THE CALCULUS 467

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FIGURE 4. Fermat's quadrature of a generalized hyperbola.

that the ratio AHm : AGm = EGn : Ç Ã is the same for any two points Å and /

on the curve; we would describe this property by saying that xmyn = const.

Powers of sines. Cavalieri found the "sums of the powers of the lines" inside a

triangle. In 1659 Pascal did the same for the "sums of the powers of the lines

inside a quadrant of a circle." Now a line inside a quadrant of a circle is what

up to now has been called a sine. Thus, Pascal found the sum of the powers

of the sines of a quadrant of a circle. In modern terms, where Cavalieri found

fix" dx = an+1/(n+l), Pascal found fP{Rsin(p)Rd(p = R(Rcosa - flcos/?).

1.3. The relation between tangents and areas. The first statement of a re-

lation between tangents and areas appears in 1670 in a book entitled Lectiones

geometricae by Isaac Barrow (1630-1677), a professor of mathematics at Cam-

bridge and later chaplain to Charles II. Barrow gave the credit for this theorem

to "that most learned man, Gregory of Aberdeen" (James Gregory, 1638-1675).

Barrow states several theorems resembling the fundamental theorem of calculus.

The first theorem (Section 11 of Lecture 10) is the easiest to understand. Given a

curve referred to an axis, Barrow constructs a second curve such that the ordinate

at each point is proportional to the area under the original curve up to that point.

We would express this relation as F(x) = (1/R) fi f(t)dt, where y = f(x) is the

first curve, y = F(x) is the second, and l/R is the constant of proportionality. If

the point Ô = (t, 0) is chosen on the axis so that (÷ — t)· f(x) = RF(x), then, said

Barrow, Ô is the foot of the subtangent to the curve y = F(x); that is, ÷ — t is

the length of the subtangent. In modern language the length of the subtangent to

the curve y = F(x) is \F(x)/F'(x)\. This expression would replace (x -1) in the

equation given by Barrow. If both F(x) and F'(x) are positive, this relation really

does say that f(x) = RF'(x) = (d/dx) fi f(t) dt.

Later, in Section 19 of Lecture 11, Barrow shows the other version of the

fundamental theorem, that is, that if a curve is chosen so that the ratio of its

ordinate to its subtangent (this ratio is precisely what we now call the derivative)

is proportional to the ordinate of a second curve, the area under the second curve

is proportional to the ordinate of the first.

1.4. Infinite series and products. The methods of integration requiring the
summing of infinitesimal rectangles or all the lines inside a plane figure led naturally
to the consideration of infinite series. Several special series were known by the
mid-seventeenth century. For example, the Scottish mathematician James Gregory