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# An Archimedean Research Theme: The Calculation of the Volume of Cylindrical Groins

01 Jan 2010-Vol. 11, pp 3-15
TL;DR: In this paper, the authors describe a method of 18th century for cilindrical groins thought by Girolamo Settimo and Nicolo di Martino, starting from Archimedes' method for calculating the volume of cylindrical wedges.
Abstract: Starting from Archimedes’ method for calculating the volume of cylindrical wedges, I want to get to describe a method of 18th century for cilindrical groins thought by Girolamo Settimo and Nicolo di Martino. Several mathematicians studied the measurement of wedges, by applying notions of infinitesimal and integral calculus; in particular I examinated Settimo’s Treatise on cylindrical groins, where the author solved several problems by means of integrals.

### INTRODUCTION

• In the Eighteenth century, several mathematicians studied the measurement of vault and cylindrical groins by means of infinitesimal and integral calculus.
• He divided both figures into infinitely many slices of infinitesimal width, and he balanced each slice of one figure against a corresponding slice of the second figure on a lever.
• Indeed, every one of these problems was concluded with integrals that were reduced to more simple integrals by means of decompositions in partial sums.

### HOW ARCHIMEDES CALCULATED THE VOLUMES OF CYLINDRICAL WEDGES

• The calculation of the volume of cylindrical wedge appears as theorem XVII of Archimedes' The Method.
• Starting from a cylinder inscribed within a prism, let us construct a wedge following the statement of Archimedes' theorem and then let us cut the prism with a plane that is perpendicular to the diameter MN (see fig 1.a), also known as It works as follows.
• The section obtained is the rectangle BAEF (see fig 1.b) , where FH' is the intersection of this new plane with the plane generating the wedge, HH'=h is the height of the cylinder and DC is the perpendicular to HH' passing through its midpoint.
• Then let us cut the prism with another plane passing through DC (see fig. 2 ).
• Besides, KL is the intersection between the two new planes that the authors constructed.

### DO : DX = H'B : H'V = BF : UV = h : u = (h•IJ) : (u•IJ).

• Then Archimedes thinks the segment CD as lever with fulcrum in O; he transposes the rectangle UVIJ at the right of the lever with arm r and the rectangle FBIJ at the left with the arm x.
• He says that it is possible to consider another segment parallel to LK, instead of IJ and the same argument is valid; therefore, the union of any rectangle like S'T'ST with arm r builds the wedge and the union of any rectangle like S'T'I'T' with arm x builds the half-cylinder.
• Then Archimedes proceeds with similar arguments in order to proof completely his theorem.
• Perhaps it is important to clarify that Archimedes works with right cylinders that have defined height and a circle as the base.

### GIROLAMO SETTIMO AND HIS HISTORICAL CONTEST

• Girolamo Settimo was born in Sicily in 1706 and studied in Palermo and in Bologna with Gabriele Manfredi (1681-1761).
• Niccolò De Martino (1701-1769) was born near Naples and was mathematician, and a diplomat.
• Their correspondence collects 62 letters of De Martino and two draft letters of Settimo; its peculiar mathematical subjects concern with methods to integrate fractional functions, resolutions of equations of any degree, method to deduce an equation of one variable from a system of two equations of two unknown quantities, methods to measure surface and volume of vaults 1 .
• Treatise on cylindrical groins that would have to contain the treatise Sulla misura delle Volte ("On the measure of vaults").
• Fitalia, and it is included in the volume Miscellanee Matematiche di Geronimo Settimo (M.SS. del sec. XVIII).

### Settimo defines cylindrical groins as follows:

• "If any cylinder is cut by a plane which intersects both its axis and its base, the part of the cylinder remaining on the base is called a cylindrical groin".
• Settimo concludes each one of these problems with integrals that are reduced to more simple integrals by means of decompositions in partial sums, solvable by means of elliptical functions, or elementary functions (polynomials, logarithms, circular arcs).
• Settimo and de Martino had consulted also Euler to solve many integrals by means of logarithms and circular arcs 2 .
• Let us examine now how Settimo solved his first problem, "How to determine volume of cylindrical groin".

### CONCLUSION

• Various authors have eredited Archimedes, but the authors know that Prof. Heiberg found the Palimpsest containing Archimedes' method only in 1907, and therefore it is practically certain that Settimo did not know Archimedes' work.
• Archimedes' solutions for calculating the volume of cylindrical wedges can be interpreted as computation of integrals, as Settimo really did, but both methods of Archimedes and Settimo are missing of generality: there is no a general computational algorithm for the calculations of volumes.
• They base the solution of each problem on a costruction determined by the special geometric features of that particular problem; Settimo however is able to take advantage of prevoious solutions of similar problems.
• It is important finally to note that Settimo, who however has studied and knew the modern infinitesimal calculus (he indeed had to consult Roger Cotes and Leonhard Euler with De Martino in order to calculate integrals by using logarithms and circular arcs), considers the construction of the infinitesimal element similarly Archimedes.
• Wanting to compare the two methods, the authors can say that both are based on geometrical constructions, from where they start to calculate infinitesimal element (that Settimo calls "elemento di solidità"): Archimedes' mechanical method was a precursor of that techniques which led to the rapid development of the calculus.

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AN ARCHIMEDEAN RESEARCH THEME: THE
CALCULATION OF THE VOLUME OF
CYLINDRICAL GROINS
Università degli Studi di Salerno
Via Ponte don Melillo, 84084 Fisciano (SA), Italy
ABSTRACT. Starting from Archimedes’ method for calculating the
volume of cylindrical wedges, I want to get to describe a method of 18th
century for cilindrical groins thought by Girolamo Settimo and Nicolò di
Martino. Several mathematicians studied the measurement of wedges, by
applying notions of infinitesimal and integral calculus; in particular I
examinated Settimo’s Treatise on cylindrical groins, where the author
solved several problems by means of integrals.
KEY WORDS: Wedge, cylindrical groin, Archimedes’ method, G.
Settimo.
INTRODUCTION
“Cylindrical groins” are general cases of cylindrical wedge, where the
base of the cylinder can be an ellipse, a parabola or a hyperbole. In the
Eighteenth century, several mathematicians studied the measurement of
vault and cylindrical groins by means of infinitesimal and integral
calculus. Also in the Kingdom of Naples, the study of these surfaces was a
topical subject until the Nineteenth century at least because a lot of public
buildings were covered with vaults of various kinds: mathematicians tried
to give answers to requirements of the civil society who vice versa
submitted concrete questions that stimulated the creation of new
procedures for extending the theoretical system.
Archimedes studied the calculation of the volume of a cylindrical wedge,
a result that reappears as theorem XVII of The Method:
If in a right prism with a parallelogram base a cylinder be inscribed
which has its bases in the opposite parallelograms [in fact squares], and
its sides [i.e., four generators] on the remaining planes (faces) of the

prism, and if through the centre of the circle which is the base of the
cylinder and (through) one side of the square in the plane opposite to it a
plane be drawn, the plane so drawn will cut off from the cylinder a
segment which is bounded by two planes, and the surface of the cylinder,
one of the two planes being the plane which has been drawn and the other
the plane in which the base of the cylinder is, and the surface being that
which is between the said planes; and the segment cut off from the cylinder
is one sixth part of the whole prism.
The method that Archimedes used for proving his theorem consist of
comparing the area or volume of a figure for which he knew the total mass
and the location of the centre of mass with the area or volume of another
figure he did not know anything about. He divided both figures into
infinitely many slices of infinitesimal width, and he balanced each slice of
one figure against a corresponding slice of the second figure on a lever.
Using this method, Archimedes was able to solve several problems that
would now be treated by integral and infinitesimal calculus.
The Palermitan mathematician Girolamo Settimo got together a part of
his studies about the theory of vaults in his Trattato delle unghiette
cilindriche (Treatise on cylindrical groins), that he wrote in 1750 about
but he never published; here the author discussed and resolved four
problems on cylindrical groins.
In his treatise, Settimo gave a significant generalization of the notion of
groin and used the actual theory of infinitesimal calculus. Indeed, every
one of these problems was concluded with integrals that were reduced to
more simple integrals by means of decompositions in partial sums.
HOW ARCHIMEDES CALCULATED THE VOLUMES OF
CYLINDRICAL WEDGES
The calculation of the volume of cylindrical wedge appears as theorem
XVII of Archimedes’ The Method. It works as follows: starting from a
cylinder inscribed within a prism, let us construct a wedge following the
statement of Archimedes theorem and then let us cut the prism with a
plane that is perpendicular to the diameter MN (see fig 1.a). The section
obtained is the rectangle BAEF (see fig 1.b), where FH’ is the intersection
of this new plane with the plane generating the wedge, HH’=h is the height
of the cylinder and DC is the perpendicular to HH’ passing through its
midpoint.
!

Fig.1.a. Construction of the
wedge.
Fig.1.b. Section of the cylinder with a
plane perpendicular to the diameter
MN.
Then let us cut the prism with another plane passing through DC (see fig.
2). The section with the prism is the square MNYZ, while the section with
the cylinder is the circle PRQR’. Besides, KL is the intersection between
the two new planes that we constructed.
!
Fig.2. Section of the cylinder with a plane passing through DC.
!
Let us draw a segment IJ parallel to LK and construct a plane through IJ
and perpendicular to RR’; this plane meets the cylinder in the rectangle

S’T’I’T’ and the wedge in the rectangle S’T’ST, as it is possible to see in
the figure 3:
Fig.3. Construction of the wedge.
Because OH’ and VU are parallel lines cut by the two transversals DO
and H’F, we have
DO : DX = H’B : H’V = BF : UV (see fig.4)
Fig.4. Sections of the wedge
where BF=h and UV is the height, u, of the rectangle S’T’ST. Therefore
DO : DX = H’B : H’V = BF : UV = h : u = (hIJ) : (uIJ).

Besides H’B=OD (that is r) and H’V=OX (that is x). Therefore
!
(FBIJ) : (UV • IJ) = r : x, and (FB IJ) • x = (UV IJ)r.
Then Archimedes thinks the segment CD as lever with fulcrum in O; he
transposes the rectangle UV•IJ at the right of the lever with arm r and the
rectangle FB•IJ at the left with the arm x. He says that it is possible to
consider another segment parallel to LK, instead of IJ and the same
argument is valid; therefore, the union of any rectangle like S’T’ST with
arm r builds the wedge and the union of any rectangle like S’T’I’T’ with
arm x builds the half-cylinder.
Then Archimedes proceeds with similar arguments in order to proof
completely his theorem.
Perhaps it is important to clarify that Archimedes works with right
cylinders that have defined height and a circle as the base.
GIROLAMO SETTIMO AND HIS HISTORICAL CONTEST
Girolamo Settimo was born in Sicily in 1706 and studied in Palermo and
in Bologna with Gabriele Manfredi (1681-1761). Niccolò De Martino
(1701-1769) was born near Naples and was mathematician, and a
diplomat. He was also one of the main exponents of the skilful group of
Italian Newtonians, whereas the Newtonianism was diffused in the
Kingdom of Naples. Settimo and De Martino met each other in Spain in
1740 and as a consequence of this occasion, when Settimo came back to
Palermo, he began an epistolar relationship with Niccolò. Their
correspondence collects 62 letters of De Martino and two draft letters of
Settimo; its peculiar mathematical subjects concern with methods to
integrate fractional functions, resolutions of equations of any degree,
method to deduce an equation of one variable from a system of two
equations of two unknown quantities, methods to measure surface and
volume of vaults
1
.
One of the most important arguments in the correspondence is also the
publication of a book of Settimo who asked De Martino to publish in
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
1
Niccolò de Martino-Girolamo Settimo. Con un saggio sull’inedito Trattato delle
Unghiette Cilindriche di Settimo, Firenze, Olschki, 2008.

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