The Garden of Archimedes
 A Museum for Mathematics

Squaring methods from antiquity to the Seventeenth Century

Theory of proportions and methods of exhaustion

One of the prime motivations that leads to the development of Mathematics is the possibility of measuring quantities, or, in other words, of linking a number to every given quantity, such that it expresses a relationship with a given sample quantity (unit of measure). Examples of this appear in all the works of the Pre-Hellenic period. The impossibility of finding a universal unit of measure that can allow all homogeneous quantities to be expressed with a whole number, and therefore the need to use submultiples, immediately leads to the introduction of fractions in the number system, even though the calculations carried out using fractions are not always fully understood or explored. Greek mathematics perfects this number system. The discovery, however, of incommensurable quantities, such as the edge of a square and its diagonal, destroys the constructs of the Pythagoreans, which placed the science of numbers at the basis of all descriptions. If numbers are inadequate for describing things - and particularly the ratios between those things - the theory of numbers becomes marginal and another theory emerges, allowing the direct use of ratios. This theory is perfected in the theory of proportions, as expounded by Euclid. From this point onwards, and in practice up to the seventeenth century, all the results regarding quantities were expressed in terms of ratios or proportions.

A class of quantities (that will be said to be homogeneous) is, from now on, clearly defined, as is known how to make a comparison between two quantities belonging to the same class (that is, how to establish which is the greater and which the smaller one of the two) and how to sum them together.

In the case of areas of plane figures or the volumes of solids, the sum is essentially their union, and the comparison is carried out by combining the criteria of inclusion with one of equivalence by equi-decomposition.

To prove that a figure $A$is equal to another figure$B$ in the case in which they are not equi-decomposable it will need to be proved that it cannot be either$A<B$ or $A>B$.

To do this, one needs to argue by contradiction. Supposing, for example, that $A<B$ one reaches the absurd by constructing an intermediate figure between$A$ and $B$ which should simultaneously be greater and smaller than $A$.

The method of exhaustion attributed to Eudoxus and used in the Twelfth book of Elements by Euclid, belongs to the classical theory of quantities. The term "exhaustion" is not used by the Greeks, but was introduced in the Sixteenth century. It refers to the procedure of constructing the intermediate figure described above based on the following axiom: if one subtracts from any given quantity a part equal to no less than half its size, and if from the remaining part, one again subtracts not less than its half, and if this process of subtraction is continued, at the end what remains is a quantity inferior to any quantity of the same kind previously assigned.

This procedure, which can be quite long and cumbersome, and requires one to know the result one wants to reach in advance, gives rigorous proof that certain figures relate to each other according to a certain relationship, or that two figures relate to each other in the same relationship of another two. So, a widespread belief developed during the Renaissance, saying that there existed a secret method that one could use without going through the whole process.

From the second half of the Sixteenth century, the problem of "divining" the so called method used in finding a short cut to all the complications that the method of exhaustion present renewed the development of new techniques, as results accompanying the discovery and restoration of the classics increased.

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|   Archimedes' quadrature of the parabola    |    Cavalieri's indivisibles    |    Torricelli's infinite hyperboloid    |   

Archimedes' quadrature of the parabola

In the introduction of the Quadrature of the parabola Archimedes declares himself the first to attempt this result and states that he can demonstrate it by using a "lemma", a variant of the axiom of exhaustion used previously by other authors, known today as the "Archimedean postulate". Given two unequal areas, it is possible to add to itself the difference by which the larger area exceeds the minor so often that any finite area will be exceeded. In proposition 17, a quite long and complex demonstration is given relating to the quadrature with a complete procedure according to the method of exhaustion. A second and simpler demonstration is contained in proposition 24, the last of the book.

To demonstrate that the segment of the parabola $ADBEC$, which, for reasons of simplicity, we shall indicate with $P$, has an area equal to $K$, which is 4/3 of the inscribed triangle $ABC$, he proves that cannot be either $P$ greater than $K$ nor $P$ smaller than $K$. In the first step this is achieved by constructing a polygon, which we shall indicate with $S$, in which case we obtain the contradiction $K<S$ and $S<K$. The polygon $S$ is constructed by the successive union of the triangle $ABC$ with the triangles inscribed in the remaining parabolic segments. In previous propositions, Archimedes proved that, by inscribing a triangle in a parabolic segment and considering the triangles inscribed in the two remaining parabolic segments, one concludes that the sum of the latter two gives an area which is equal to a quarter of the first triangle. If, therefore, one takes $P>K$ , by repeating the construction, one can form a polygon $S$ such that $P>S>K$. On the other hand, we get $S<K$ (note that 1+(1/4)+(1/4)2+(1/4)3+...=4/3) and thus the first contradiction is achieved. In analogous fashion, one can proceed in the second step to prove that cannot be less than $P$ $K$.

* Exhibit I.1

Quadratura parabolae, from the translation by Francisci Maurolici
Proposition 24.
The area of any section of a right cone is equal to 4/3 of the triangle which has the same base as the segment and equal height.
Note that the section is understood as being perpendicular to the axis of the cone. The figure thus obtained is a parabola. In the version of Archimedes' works edited by Maurolico, which differs in several parts from the classical version, the final proposition XXIV is numbered as the XXV. The quadrature of the parabola is also the first example to be illustrated in the letter by Eratosthenes, as by J.L. Heiberg (The Method, proposition 1) discovered in 1906. Here, Archimedes ponders on equilibrium using concepts of mechanical theory, applying the properties of centres of gravity and levers to unveil theorems through mechanics. This heuristic method of supposing results, does not give, as Archimedes himself stated, a real demonstration, but is simply an indication as to the conclusion that can be reached through a geometrical demonstration.
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The indivisibles of Cavalieri

Unlike the traditional methods, the method of indivisibles of Cavalieri presents itself as a new and powerful tool for the determination of areas and volumes. In the introduction of Geometria he describes how he came to his elaboration:
Whilst meditating one day on the generation of solids that originate from a rotation around an axis, and comparing the ratio of the generating plane figures and that of the solids generated, I was amazed by the fact that the figures generated were so far removed from the initial conditions of their parent figures, that they proved to be following a completely different ratio from theirs. For example, a cylinder obtained in a similar fashion as a cone with equal base, through a rotation round the same axis, is three times the cone, even if generated by the revolution of a parallelogram which is two times the triangle that generates the cone. [...]
Having thus more than once drawn attention to this difference in many other figures, whilst previously imagining, for example, a cylinder as the union of an indefinite number of parallelograms or a cone having equal base and height, as the union of an indefinite number of triangles all passing through the axis, I believed that by obtaining the mutual ratio of these plane figures, the ratio of the solids they generated would immediately become apparent, and, however, it being very clear that the ratio of the generating plane figures did not concord at all with that of the solids generated, it seemed to me that one should rightly conclude that one would have lost time and effort and, that one would have reaped useless hay, if one would have endeavoured in the search of the measure of figures by adopting such a method.
But after having considered the issue more deeply, I finally came to the following opinion, precisely that, for our purpose, one should not take intersecting planes but planes that are parallel. Following this suggestion, I investigated many cases, and in all I found a perfect correspondence between both in the ratio of the solids and that of their plane sections, and also in the ratio of the planes and that of their lines [...].
Having thus considered the said cylinder and cone as intersected no longer by the axis but rather, parallel to the base, I discovered that the planes I define in book II as "all planes" of the cylinder and "all planes" of the cone, have a ratio equal to that of the cylinder to the cone, when referred to the same base [...]. I therefore considered the method of examining the ratios between lines instead of those between planes, and the ratios between planes instead of those between solids, an excellent method for investigating figures, so that I could obtain a quick measure of the figures themselves. The thing went, I believe, according to my wishes, as will become clear to those who read it all.
In the first and second book he expounds the "lemmatic propositions", or in other words, the lemmas on which his method was based, introducing the concept of "all the lines" of a plane and "all the planes" of a solid figure and establishing that "all the lines" of plane figures (and similarly "all planes of solid figures") are quantities that are related to each other with a ratio, a fundamental outcome for those who wanted to work with such figures. In the books that followed he gave a demonstration of the results relating to plane and solid figures, generated by conic sections and spirals. The method of indivisibles was much attacked. Cavalieri tried to anticipate this criticism in the Geometria itself: in book VII he expounds that which he calls the "second method" in which he gives a clarification of the foundations of the indivisibles taking basis from theorem I, still known today as the "principle of Cavalieri".

* Exhibit I.2

Bonaventura Cavalieri
Geometria indivisibilibus quadam ratione promota
Theorem I. Proposition I.
Any given plane figures, placed in between the same parallels, are said to be equal if, for any straight line drawn equidistant to the said parallels, the portions thus intersected of these lines are equal.
And any given solid figures placed in between the same parallel planes, in which for any planes drawn equidistantly to the parallel planes in question, the plane figures thus generated in the solids are equal, these are to be considered equal too.
Let us thus call these figures, both plane and solid compared to each other and in relation to the lines of reference or to the parallel planes between which they are placed if it is necessary to give an indication, as "equally similar".
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The infinite hyperboloid of Torricelli

One of the results that Torricelli achieved through the application of the method of indivisibles, which more than others earned the admiration of his contemporaries, was the calculus of the volume of an hyperbolic solid. This problem "would seem not only difficult but actually impossible to any aspiring geometer", he writes in his introduction and continues:

In fact in the scholarly tracts on Geometry, one finds measures of figures which are limited on all sides and [...] none of them, as far as I know have an infinite extension. And if one proposes to consider a solid or an infinitely extended plane figure, everyone immediately conceives a figure of this sort as being of infinite quantity. The solid of infinite quantity does exist, of course, but it so fine in thickness that no matter how prolonged towards infinite, it is no larger in quantity than a small cylinder. It is a solid generated by the hyperbola [...]
Which Torricelli also calls "acute hyperbolic solid".

Cavalieri himself was surprised by such an achievement and in a letter to Torricelli he writes:

I do not know how you have managed so easily to pick out of the infinite depth of that solid its true dimension, since to me it seems infinitely long.
The proofs given by Torricelli are two: together with the one using indivisibles, there is another proof that will satisfy even the "the reader who is not on good terms with indivisibles" [...] "and that is the usual method of demonstration of the ancient geometers, which is much longer, but not for this reason, in my opinion, much more certain". The introduction contains a eulogy of the Geometry of indivisibles, "which is a direct, scientific, and so to say, natural manner of giving proof", and following the enthusiasm about horizons opened by the new discovery, Torricelli stated:
The old geometry makes me feel compassion, because by ignoring or not wanting to recognise the existence of indivisibles, in the study of solid bodies, it discovered such few truths that a painful lack of ideas has persisted down to our age. In fact, the theorems of the ancients, that are part and parcel of the doctrine of solids, represent only a part of the speculations, that in our time the admirable Cavalieri, not to speak of others, have elaborated, in many different kinds and in great number, concerning many classes of solids."

* Exhibit I. 3

Evangelista Torricelli
On the measure of the parabola and the hyperbolic solid
The infinitely long acute hyperbolic solid, intersected with a plane perpendicular to the axis, is equal, together with the cylinder of its base, to the right cylinder having the converse edge, or the axis of the hyperbola, as base, and height equal to the semi-diameter of the base of the acute solid.
Torricelli therefore sustains that the solid FENBLD obtained from the rotation around the axis BA is equal to the cylinder having AC as height and a circle with AH as diameter, as base, AH being the convex edge of the hyperbola. The method used by Torricelli is based on curved indivisibles, which are illustrated previously by means of a few examples " not having Cavalieri made any reference to such things, in expounding his geometry". Curved indivisibles used for the paraboloid are cylindrical surfaces described by the rotation of any segment joining a point on the hyperbole with the axis AC, as given in the figure IL. According to the properties of the hyperbola, as the point I varies, the cylindrical surfaces ONLI will be equal to the circles with diameter equal to IM, obtained by the plane intersections of the cylinder ACGH. Therefore, using the theory of indivisibles, all the cylindrical surfaces taken together will be equal to all the circles together, or in other words, the acute hyperbolic solid will be equal to the cylinder.
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