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 Guldin’s 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 {!) 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 ratio 
compounded {!) of the ratio 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 Pote- 
statem Rotundam uno grado altiorem Potestate sive Quantitate 
Rotata O 
Pappus adds that 
‘ 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.’ 
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 pro 
position. 
1 Centroharyca, Lib. ii, chap, viii, Prop. 3. Yiennae 1641. 
D d 2