Pappus's centroid theorem
The theorems are attributed to Pappus of Alexandria[a] and Paul Guldin.
[b] Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.
[4] The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by the geometric centroid of C:
For example, the surface area of the torus with minor radius r and major radius R is
A curve given by the positive function
is an infinitesimal line element tangent to the curve, the length of the curve is given by:
component of the centroid of this curve is:
The area of the surface generated by rotating the curve around the x-axis is given by:
Using the last two equations to eliminate the integral we have:
The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F. (The centroid of F is usually different from the centroid of its boundary curve C.) That is:
This special case was derived by Johannes Kepler using infinitesimals.
[c] The area bounded by the two functions:
component of the centroid of this area is given by:
If this area is rotated about the y-axis, the volume generated can be calculated using the shell method.
Using the last two equations to eliminate the integral we have:
is given by the change of variables formula:
is the determinant of the Jacobian matrix of the change of variables.
The last equality holds because the axis of rotation must be external to
The theorems can be generalized for arbitrary curves and shapes, under appropriate conditions.
If the figure F moves through space so that it remains perpendicular to the curve L traced by the centroid of F, then it sweeps out a solid of volume V = Ad, where A is the area of F and d is the length of L. (This assumes the solid does not intersect itself.)
In particular, F may rotate about its centroid during the motion.
However, the corresponding generalization of the first theorem is only true if the curve L traced by the centroid lies in a plane perpendicular to the plane of C. In general, one can generate an
dimensional solid by rotating an
Then Pappus' theorems generalize to:[7] Volume of
-th centroid of the generating solid) and Surface area of
= (Surface area of generating
-th centroid of the generating solid) The original theorems are the case with
When I see everyone occupied with the rudiments of mathematics and of the material for inquiries that nature sets before us, I am ashamed; I for one have proved things that are much more valuable and offer much application.
In order not to end my discourse declaiming this with empty hands, I will give this for the benefit of the readers: The ratio of solids of complete revolution is compounded of (that) of the revolved figures and (that) of the straight lines similarly drawn to the axes from the centers of gravity in them; that of (solids of) incomplete (revolution) from (that) of the revolved figures and (that) of the arcs that the centers of gravity in them describe, where the (ratio) of these arcs is, of course, (compounded) of (that) of the (lines) drawn and (that) of the angles of revolution that their extremities contain, if these (lines) are also at (right angles) to the axes.
These propositions, which are practically a single one, contain many theorems of all kinds, for curves and surfaces and solids, all at once and by one proof, things not yet and things already demonstrated, such as those in the twelfth book of the First Elements.
