substitution. So, with the indefinite integral:
Now consider the remaining integral and use the substitution u = x 2 , in which case d u /d x = 2 x . We then have
Putting all this together results in the surface area of the trumpet being given by
As N → ∞ the first term clearly approachesbut the log function increases without bound, which means that the surface area also increases without bound.
(It is appropriate that an anagram of ‘Evangelista Torricelli’ is ‘Lo! It is a clever integral’.)
The Trumpet’s Centre of Mass
The confusion is complete when we consider a comment of Wal-lis that a
surface, or solid, may be supposed to be so constituted, as to be Infinitely Long, but Finitely Great, (the Breadthcontinually decreasing in greater proportion than the Length Increaseth,) and so as to have no Centre of Gravity. Such is Toricellio’s Solidum Hyperbolicum acutum.
Using the standard calculus definition of the centre of massof a solid of revolution about the x -axis, we have that
and we have in our case
A Drinking Vessel
So, in 1643 Torricelli brought to the mathematical world a solid which has infinite surface area but finite volume. In 1658 Christiaan Huygens and René François de Sluze added to the mathematical unease of the time by reversing the conditions: their solid has finite surface area and infinite volume.
We will not consider their arguments or more modern ones to establish the fact, but the solid is generated from the cissoid (meaning ‘ivy-shaped’). The canonical curve has equation y 2 = x 3 /(1 − x ), which is shown in figure 8.5 ; evidently, it has a vertical asymptote at x = 1. It is attributed to Diocles in about 180 bc in connection with his attempt to duplicate the cube by geometrical methods and appears in Eutocius’s commentaries on Archimedes’ On the Sphere and the Cylinder , wherein the method of exhaustion was developed. The solid concerned is contained between the rotation of the upper half of the cissoid and the vertical asymptote about the y -axis; it forms a goblet-shaped figure, as shown in figure 8.6 .
Figure 8.5.
Figure 8.6. The cissoid.
In a letter to Huygens, de Sluze mischievously described the solid as
a drinking glass that had small weight, but that even the hardiest drinker could not empty
(levi opera deducitur mensura vasculi, pondere non magni, quod interim helluo nullus ebibat).
Torricelli’s Trumpet would satisfy the more moderate drinker, but the glass could never be wetted! Admittedly, this is fancifulfor several important reasons, but the imagery is compelling. Where is the paradox? As ever, our senses have deceived us when we have brought about the confusion which arises when we try to bring to the real world something which cannot exist within it; infinitely long things cannot be brought into reality (Euclid’s parallels postulate reveals the danger in trying to do so) and wine is not infinitely thin.
Galileo’s own view echoes this:
[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited.
But we should leave the last word to Hobbes, when he commented on the assertion that an infinite solid of finite volume exists:
To understand this for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad.
Chapter 9
NONTRANSITIVE EFFECTS
We want the surprise to be transitive like the impatient thump which unexpectedly restores the picture to the television set, or the electric shock which sets the fibrillating heart back to its proper rhythm.
Seamus Heaney
The Background
A dictionary definition of the adjective ‘transitive’ is ‘being or relating to a relationship with the property that if the relationship holds between a first element and a second and between the second element and a third, it holds between the first and third elements.’
Initially it is easy
Now consider the remaining integral and use the substitution u = x 2 , in which case d u /d x = 2 x . We then have
Putting all this together results in the surface area of the trumpet being given by
As N → ∞ the first term clearly approachesbut the log function increases without bound, which means that the surface area also increases without bound.
(It is appropriate that an anagram of ‘Evangelista Torricelli’ is ‘Lo! It is a clever integral’.)
The Trumpet’s Centre of Mass
The confusion is complete when we consider a comment of Wal-lis that a
surface, or solid, may be supposed to be so constituted, as to be Infinitely Long, but Finitely Great, (the Breadthcontinually decreasing in greater proportion than the Length Increaseth,) and so as to have no Centre of Gravity. Such is Toricellio’s Solidum Hyperbolicum acutum.
Using the standard calculus definition of the centre of massof a solid of revolution about the x -axis, we have that
and we have in our case
A Drinking Vessel
So, in 1643 Torricelli brought to the mathematical world a solid which has infinite surface area but finite volume. In 1658 Christiaan Huygens and René François de Sluze added to the mathematical unease of the time by reversing the conditions: their solid has finite surface area and infinite volume.
We will not consider their arguments or more modern ones to establish the fact, but the solid is generated from the cissoid (meaning ‘ivy-shaped’). The canonical curve has equation y 2 = x 3 /(1 − x ), which is shown in figure 8.5 ; evidently, it has a vertical asymptote at x = 1. It is attributed to Diocles in about 180 bc in connection with his attempt to duplicate the cube by geometrical methods and appears in Eutocius’s commentaries on Archimedes’ On the Sphere and the Cylinder , wherein the method of exhaustion was developed. The solid concerned is contained between the rotation of the upper half of the cissoid and the vertical asymptote about the y -axis; it forms a goblet-shaped figure, as shown in figure 8.6 .
Figure 8.5.
Figure 8.6. The cissoid.
In a letter to Huygens, de Sluze mischievously described the solid as
a drinking glass that had small weight, but that even the hardiest drinker could not empty
(levi opera deducitur mensura vasculi, pondere non magni, quod interim helluo nullus ebibat).
Torricelli’s Trumpet would satisfy the more moderate drinker, but the glass could never be wetted! Admittedly, this is fancifulfor several important reasons, but the imagery is compelling. Where is the paradox? As ever, our senses have deceived us when we have brought about the confusion which arises when we try to bring to the real world something which cannot exist within it; infinitely long things cannot be brought into reality (Euclid’s parallels postulate reveals the danger in trying to do so) and wine is not infinitely thin.
Galileo’s own view echoes this:
[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited.
But we should leave the last word to Hobbes, when he commented on the assertion that an infinite solid of finite volume exists:
To understand this for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad.
Chapter 9
NONTRANSITIVE EFFECTS
We want the surprise to be transitive like the impatient thump which unexpectedly restores the picture to the television set, or the electric shock which sets the fibrillating heart back to its proper rhythm.
Seamus Heaney
The Background
A dictionary definition of the adjective ‘transitive’ is ‘being or relating to a relationship with the property that if the relationship holds between a first element and a second and between the second element and a third, it holds between the first and third elements.’
Initially it is easy
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