AN ARCHIMEDEAN RESEARCH THEME: THE

CALCULATION OF THE VOLUME OF

CYLINDRICAL GROINS

Nicla Palladino

Università degli Studi di Salerno

Via Ponte don Melillo, 84084 Fisciano (SA), Italy

e-mail: nicla.palladino@unina.it

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 = (h•IJ) : (u•IJ).

Besides H’B=OD (that is r) and H’V=OX (that is x). Therefore

!

(FB • IJ) : (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

N. Palladino - A.M. Mercurio - F. Palladino, La corrispondenza epistolare

Niccolò de Martino-Girolamo Settimo. Con un saggio sull’inedito Trattato delle

Unghiette Cilindriche di Settimo, Firenze, Olschki, 2008.