An Archimedean Research Theme: The Calculation of the Volume of Cylindrical Groins
Summary (2 min read)
- 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".
- 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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Q1. What are the contributions mentioned in the paper "An archimedean research theme: the calculation of the volume of cylindrical groins" ?
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.