A history of Greek mathematics - Wilbourhall.org

THE COLLECTION. BOOK VII 403
utility. In justification of this statement and
'
in order that
he may not appear empty-handed when leaving the subject ',
he will present his readers with the following.
(Anticipation of Guldins Theorem,)
The enunciations are not very clearly worded, but there
is no doubt as to the sense.
Figures generated by a complete revolution of a plane figure
'
about an axis are in a ratio compounded (1) of the ratio
of the areas of the figures, and (2) of the ratio of the straight
lines similarly drawn to (i.e. drawn to meet at the same angles)
the axes of rotation from the respective centres of gravity.
Figures generated by incomplete revolutions are in the rcdio
compounded (1) of the rcdio of the areas of the figures and
(2) of the ratio of the arcs described by the centres of gravity
of the respective figures, the latter rcdio being itself compounded
(a) of the ratio of the straight lines similarly drawn {from
the respective centres of gravity to the axes of rotation) and
(b) of the ratio of the angles contained (i. e. described) about
the axes of revolution by the extremities of the said straight
lines (i.e.
the centres of gravity).'
Here, obviously, we have the essence of the celebrated
theorem commonly attributed to P. Guldin (1577-1643),
'
quantitas rotunda in viam rotationis ducta producit Potestatem
Rotundam uno grado altiorem Potestate sive Quantitate
Rotata *}
Pappus adds that
c
these propositions, which are practically one, include any
number of theorems of all sorts about curves, surfaces, and
solids, all of which are proved at once by one demonstration,
and include propositions both old and new, and in particular
those proved in the twelfth Book of these Elements. 5
Hultsch attributes the whole passage (pp. 680. 30-682. 20)
to an interpolator, I do not know for what reason; but it
seems to me that the propositions are quite beyond what
could be expected from an interpolator, indeed I know of
no Greek mathematician from Pappus's day onward except
Pappus himself who was capable of discovering such a proposition.
1
Centrobaryca, Lib. ii, chap, viii, Prop. 3. Viemiae 1641.
Dd2