Katz and Sherry’s [Pt.4.5] “Leibniz’ Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond,” 4.5 ‘Mathematical Implementation of Status Transitus,’ summary

summary by Corry Shores
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Mikhail G. Katz  and David Sherry

“Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”

4. Cum Prodiisset

 

4.5 Mathematical Implementation of Status Transitus

Brief Summary:

In Cum Prodiisset Leibniz discusses his Law of Continuity, and he provides some examples. Katz and Sherry will examine some of the mathematical procedures involved in these examples. What we find is that the same formulations and procedures are used regardless of whether we are dealing with finite or infinite values. This is because the law of continuity postulates that what holds for the finite holds for the infinite (and infinitesimal), [because there is a continuous transition from one to the other.]

Summary

 

In section 4.2 and 4.4, Katz and Sherry (KS) discuss three applications of Leibniz’ law of continuity and infinitesimals in his Cum Prodiisset. Now they will examine them more mathematically.

Leibniz distinguishes assignable finite quantity d(x) from infinitesimal value dx.

The assignable quantity (d)x passes via infinitesimal dx on its way to absolute 0. Then the infinitesimal dx is the terminus, or the status transitus. Zero is merely the shadow of the infinitesimal. This particular status transitus is the foundation rock of the Leibnizian definition of the differential quotient.
[KS 581]

[Recall Leibniz’ example of finding the tangent to a parabola. He writes:

“let ₁X₂X, the difference between A₁X and A₂X, be called dx;”

“and similarly, let D₂Y, the difference between ₁X₁Y and ₂X₂Y, be called dy.” (Leibniz 151)

Then, we shrink dx and dy down to find the tangent.

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

Here Leibniz does not notate a difference between dx and d(x), but we would think of the finitely large line as being d(x) and the infinitesimally small one as dx.

KS also refer to Z as being the shadow of the infinitesimal. In the appendix they explain the shadow as being a real number assigned to hyperreal infinitesimals. It is the next closest value, only infinitesimally different. In these cases below, the shadow is like rounding the infinitesimal value to the nearest real number.]]

Let’s recall the first example Katz and Sherry examine:

(1) In the context of a discussion of parallel lines, he writes: when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small (Child 1920, p. 148).
[[KS579]]

[Image from Leibniz/Child 148]

[We animated it thus:

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

]

In KS’s rendition, they have us think of this diagram on its side with the moving line intercepting the x axis. In their formulation, H is the x-intercept value.

Example (1) of parallel lines can be elaborated as follows. Let’s follow Leibniz in building the line through the point (0,1) parallel to the x-axis in the plane. Line L_H with y-intercept 1 and x-intercept H is given by y = x / H .
[KS 581]

[Referring back to Leibniz’ diagram, the y-intercept here is like point P, which remains the same, all while the x-intercept, equivalent to point C, moves to greater distances.]

We will move the x-intercept to infinity. [KS later refer to the ‘finite portion of L_H.’ Perhaps one way to picture this is to think that the lines meet way off into infinity, but they ‘drag’ the finite part of the line all the way up such that it is only infinitesimally away from being y = 1. Yet, somewhere the lines must meet, which means it cannot stay at y = 1 everywhere. So does that mean that in the infinite part, it is no longer infinitely close to being parallel? I am not sure, but I think it is always infinitely close to being parallel, but just not always being infinitely close to y = 1. So far off into infinity, it will maybe be infinitely close to being parallel to y = 0.5, and so on diminishing. For every y position of the line in its infinite part, there would be an infinite stretch both ways, making every position be nearly parallel to the line.]

Now let H be infinite. The resulting line LH has negative infinitesimal slope, meets the x-axis at an infinite point, and forms an infinitesimal angle with the x-axis at the point where they meet. We will denote by st(x) the assignable (i.e., real) shadow of a finite x. [KS 581]

[They now will formulate for finite values for the x and y locations included in Line H.]

Then every finite point (x,y) ∈ L_H satisfies

[by substituting the formula for y. Now, because we are supposing H to be infinite, that means you have a finite value divided by an infinite one, which means it is nearly zero, which means that the y value is very close to 1, and thus:]

Hence the finite portion of L_H is infinitely close to the line y = 1. The line y = 1 is parallel to the x-axis, and is merely the shadow of the inassignable L_H. Thus, the parallel line is constructed by varying the oblique line depending on a parameter. Such variation comprises the status transitus L_H defined by an infinite value of H.
[KS 582]

Now recall example 3:

(3) Finally, a conception of a parabola expressed by means of an ellipse with an infinitely removed focal point is articulated in the following terms: a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex (Child 1920, p. 148).
[KS 579]

[[Recall from our prior discussion that by moving one focus of the ellipse infinitely away from the other, the ellipse transforms into a parabola.

(Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez)

]

KS will provide a formulation to illustrate this better.

To implement example (3), let’s follow Leibniz in deforming an ellipse, via a status transitus, into a parabola. The ellipse with vertex (apex) at (0,-1) and with foci at the origin and at (0,H) is given by

[They then square both sides and expand the values:]

Then they kept the radical on the left side, moving the rest over:

Then they square both sides, and reduce to get:

[We notice that even though H is an infinite number, they still apply operations on the figures that normally apply to finite numbers. But whether or not that is mathematically admissible is uncertain, hence it the ‘postulate’ of the law of continuity that we can treat them in like manner.]

The calculation (4.1) through (4.4) depends on the following habits of general reasoning (to echo Child’s translation) with assignable quantities, which are generalized to apply to inassignable quantities (such as the terminus/status transitus) in accordance with the law of continuity:

• squaring undoes a radical;

• the binomial formula;

• terms in an equation can be transferred to the other side; etc.

General reasoning of this type is familiar in the realm of ordinary finite real numbers, but why does it remain valid when applied to the realm of infinite or infinitesimal numbers? The validity of transfering such general reasoning originally instituted in the finite realm, to the realm of the infinite is postulated by Leibniz’s law of continuity.
[KS 582]

[Because the same general reasoning is used for finite and infinite, that might give a ‘between’ status or ambiguous status to the status transitus. Notice that we are still using a formulation for an ellipse. And yet, the resulting figure is in transit to being a parabola. Off in infinity, it might be still elliptical. But it draws out the finite part such that only parabolic parts remain within the realm of assignable values.]

We therefore apply Leibniz’s law of continuity to Eq. (4.4) for an infinite H. The resulting entity is still an ellipse of sorts, to the extent that it satisfies all of the Eqs. (4.1) to (4.4). However, this entity is no longer finite. It represents a Leibnizian status transitus between ellipse and parabola. This status transitus has foci at the | origin and at an infinitely distant point (0, H).
[KS 582|583, boldface mine]

[[Recall that we left off before with:

It seems in the following that KS will assign real numbers so to find the “real shadow” of this figure, that is, the finite assignable values it is infinitely close to.]]

Assuming x and y are finite, we set x₀ = st(x) and y₀ = st(y), to obtain a real shadow of this entity:

then simplifying:

Thus, the finite portion of the status transitus (4.4) is infinitely close to its shadow (4.5), namely the real parabola y ¼ x2 4 1 (in Leibniz’s terminology as translated by Child, ‘‘it is really true’’ that this parabola has no focus at infinity—see Sect. 4.4). This is the kind of payoff Leibniz is seeking with his law of continuity.
[KS 583]

Bibliography:

Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Examples of Status Transitus / Status Terminus and the Law of Continuity in Leibniz’ Cum Prodiisset

summary by Corry Shores
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Leibniz

 

Examples of Status Transitus / Status Terminus and the Law of Continuity in

 Cum Prodiisset

 

Brief Summary:

In Cum Prodiisset Leibniz discusses his Law of Continuity. According to one formulation, in a continuous transition, the final ending (the terminus) of the transition may be included with that transition. He provides a number of examples to illustrate. They show changes from opposing states happening continuously and by means of an infinitesimally small variation, for example, unequal and equal, motion and rest, convergent and parallel, and enclosed and open. On account of the law of continuity, the same calculatory procedures apply both when the values are finite and also as they pass into the infinite.

Summary

 

Before giving his examples of status transitus [/status terminus] (a concept bound up with his law of continuity), he first formulates it thus:

In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included. [Leibniz 147]

 

[[See especially Katz and Sherry’s discussion of this text, section 4 of “Leibniz’ Infinitesimals”]] Leibniz will repeat this idea of ‘a general reasoning’ for his examples. One of them involves calculations, so we will skip first to that one to gain a more technical understanding of the term. [The following, up to ‘Example 1’ is quotation except for material in double brackets. The main idea we will obtain is that a general reasoning allows the same calculating procedures to continue even after values shift from finite to infinitesimal.]

“

First of all, the sense in which the phrase "dy is the element of y," is to be taken will best be understood by considering a line A  referred to a straight line AX as axis.

Let the curve AY be a parabola,

and let the tangent at the vertex A be taken as the axis. [[Note, the Latin original indicates the axis as AX: et assumtos Axis AX sit tangens parabolae in vertice A. (p.44)]]

If AX is called x,

and AY, y,

[[given the way that dy is defined below, y could instead be the distance from A to Y, when Y is understood as a vertical axis.

]]

and the latus-rectum is a [[the latus-rectum is often defined as the line going through the focus, perpendicular to the axis, with endpoints on the parabola. It does not seem to be shown here]], the equation to the parabola will be xx = ay, and this holds good at every point. Now, let A₁X= x,

and ₁X₁Y = y

and from the point ₁Y let fall a perpendicular ₁YD to some greater ordinate ₂X₂Y that follows,

and let ₁X₂X, the difference between A₁X and A₂X, be called dx;

and similarly, let D₂Y, the difference between ₁X₁Y and ₂X₂Y, be called dy.

Then, since y = xx: a, by the same law,

[[

xx = ay
xx / a = ay / a
xx /a = y
y = xx / a

]]

we have

y+ dy = xx + 2xdx + dxdx, : a ;

[[regarding the above: since we are adding dy to y, perhaps then the other side needs to be (x + dx)(x + dx), because we would be adding dx to both x’s just as we add dy to all cases of y. Then we would obtain the above formulation by expanding it.]]

and taking away the y from the one side and the xx: a from the other, we have left

dy: dx = 2x + dx : a ;

[[It seems we can remove the y = xx / a, perhaps because the same formulation repeats with the dy and dx. That leaves us with:

dy = (2xdx + dxdx) / a ;

Then we move a dx to the other side:

dy / dx = (2xdx + dxdx) / dx(a)

Producing the formula again:

dy / dx = 2x + dx / a

]]

and this is a general rule, expressing the ratio of the difference of the ordinates to the difference of the abscissae, [[the formulation y = xx / a gave us the relation of the ordinates to the abscissae. Then we reformulated it to give us the relation between the differences of the abscissae and ordinate by means of dy and dx.]]

or, if the chord ₁Y₂Y is produced until it meets the axis in T,

then the ratio of the ordinate ₁X₁Y to T₁X, the part of the axis intercepted between the point of intersection and the ordinate, will be as 2x+ dx to a. Now, since by our postulate it is permissible to include under the one general reasoning the case also in which the ordinate ₂X ₂Y is moved up nearer and nearer to the fixed ordinate ₁X ₁Y until it ultimately coincides with it, it is evident that in this case dx becomes equal to zero and should be neglected, and thus it is clear that, since in this case T₁Y is the tangent, ₁X₁Y is to T₁X as 2x is to a.

[[Leibniz then seems to state that what he means with the law of continuity is that we can regard as equivalent cases both with evanescent values and with them removed.]]

Hence, it may be seen that there is no need in the whole of our differential calculus to say that those things are equal which have a difference that is infinitely small, but that those things can be taken as equal that have not any difference at all, provided that the calculation is supposed to be general, including both the cases | in which there is a difference and in which the difference is zero; and provided that the difference is not assumed to be zero until the calculation is purged as far as is possible by legitimate omissions, and reduced to ratios of non-evanescent quantities, and we finally come to the point where we apply our result to the ultimate case.

[[Thus to ‘include under one general reasoning seems to mean in this case that we consider the formula for the parabola as holding for all values of ₁Y₂Y, including when that value vanishes to zero. Perhaps then in the examples that precede this one in the text, which we will discuss below, ‘under one general reasoning’ might as well mean that whatever formula we regard as applying in the finite cases as well apply in the infinitesimal case.]]

” [The above quoted material from Leibniz 151-152]

Example 1:

We consider quantities A and B, where B is larger than A. However, A is diminishing until equaling B. Nonetheless, we can include its value at B as belonging in unbroken continuum with the prior values, even though their states are contrary [[when vanishing to B, A is both equal and unequal to B]]. Here is the passage as quotation:

if A and B are any two quantities, of which the former is the greater and the latter is the less, and while B remains the same, it is supposed that A is continually diminished, until A becomes equal to B ; then it will be permissible to include under a general reasoning the prior cases in which A was greater than B, and also the ultimate case in which the difference vanishes and A is equal to B. [Leibniz 147]

Example 2:

Consider two bodies in motion, A and B. A’s velocity is continuously diminishing to zero (rest) all while B’s remains the same. Even though A’s motion is coming to a rest, we can include all its variations in speed, including its coming-to-rest state, with B’s motion. [[See quoted text below. Perhaps we are to think of A and B as beginning at the same speed, so to both illustrate the continuity of change with the dichotomy of states, motion and rest]]

Similarly, if two bodies are in motion at the same time, and it is assumed that while the motion of B remains the same, the velocity of A is continually diminished until it vanishes altogether, or the speed of A becomes zero ; it will be permissible to include this case with the case of the motion of B under one general reasoning. [Leibniz 147]

Example 3:

Consider of we have a line converging with another at some angle. We pivot the line on some fixed point (P above), extending the line so that it continues to converge with the other line. The angle continuously diminishes. When it reaches the infinitely small, the lines are becoming parallel. Even though the line’s position is discontinuous, this transition is included with all prior ones.

[Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez]

From the translation:

We do the same thing in geometry, when two straight lines are taken, produced in any manner, one VA being given in position or remaining in the same site, the other BP passing through a given point P, and varying in position while the point P remains fixed ; at first indeed converging toward the line VA and meeting it in the point C ; then, as the angle of inclination VCA is continually diminished, meeting VA in some more remote point (C), until at length from BP, through the position (B)P, it comes | to βP, in which the straight line no longer converges toward VA, but is parallel to it, and C is an impossible or imaginary point. With this supposition it is permissible to include under some one general reasoning not only all the intermediate cases such as (B)P but also the ultimate case βP. [Leibniz trans 147. Above case of  “as the angle of inclination VCA is continually diminished” should have BCA instead, “deinde si angulus inclinationis, ut BCA continue minuatur” ]

Example 4:

Consider an ellipse. One focus moves away from other. In the infinitesimally small movement from finite to infinitely far, the figure changes from ellipse to parabola, and thus from enclosed to open.

[Animated diagram by Corry Shores, using OpenOffice Draw and Unfreez]

From the text:

Hence also it comes to pass that we include as one case ellipses and the parabola, just as if A is considered to be one focus of an ellipse (of which V is the given vertex), and this focus remains fixed, while the other focus is variable as we pass from ellipse to ellipse, until at length (in the case when the line BP, by its intersection with the line VA, gives the variable focus) the focus C becomes evanescent73 or impossible, in which case the ellipse passes into a parabola. Hence it is permissible with our postulate that a parabola should be considered with ellipses under a common reasoning. [ft 73: “ The term is here used with the idea of "vanishing into the far distance." ”]Just as it is common practice to make use of this method in geometrical constructions, when they include under one general construction many different cases, noting that in a certain case the converging straight line passes into a parallel straight line, the angle between it and another straight line vanishing. [Leibniz 148]

Leibniz continues, refering to these examples:

Moreover, from this postulate arise certain expressions which are generally used for the sake of convenience, but seem to contain an absurdity, although it is one that causes no hindrance, when its proper meaning is substituted. For instance, we speak of an imaginary point of intersection as if it were a real point, in the same manner as · in algebra imaginary roots are considered as accepted numbers. Hence, preserving the analogy, we say that, when the straight line BP ultimately becomes parallel to the straight line VA, even then it converges toward it or makes an angle with it, only that the angle is then infinitely small; similarly, when a body ultimately comes to rest, it is still said to have a velocity, but one that is infinitely small ; and, when one straight line is equal to another, it is said to be unequal to it, but that the difference is infinitely small ; and that a parabola is the ultimate form of an ellipse, in which the second focus is at an infinite distance from the given focus nearest to the given vertex, or in which the ratio of PA to AC, or the angle BCA, is infinitely small. [Leibniz 148]

Bibliography:

English references from:

Leibniz. The Early Mathematical Manuscripts of Leibniz. Trans. J.M. Child. Mineola, NY: Dover, 2005 [1920 Open Court].

1920 Edition available at archive.org:

https://archive.org/details/earlymathematic01gerhgoog

Latin references from:

Leibniz. Historia et origo calculi differentialis. Ed. C.I. Gerhardt. Hannover: Im Verlage der Hahn'schen Hofbuchhandlung, 1846]

Available at archive.org:

https://archive.org/details/historiaetorigo00gerhgoog

The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.

http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Archimedes in Boyer’s History of the Calculus, 1, counterbalanced parabola; 2, parabola exhaustion

by Corry Shores
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Carl Boyer

The History of the Calculus and Its Conceptual Development (The Concept of the Calculus)

Chapter 1

Selection, first two Archimedes demonstrations (pp. 48-53)

[The following is quotation, except for material between double brackets]

The greatest mathematician of antiquity, Archimedes of Syracuse, displayed two natures, for he tempered the strong transcendental imagination of Plato with the meticulously correct procedure of Euclid. He "gave birth to the calculus of the infinite conceived and brought to perfection successively by Kepler, Cavalieri, Fermat, Leibniz, and Newton,"¹³⁴  [ft 134: Chasles, A perçu historique sur l’origine et le développement des méthods en géométrie, p.22.] and so made the concepts of the derivative and the integral possible. In the demonstration of his results, however, he adhered to the clearly visualized details of the Eudoxian procedure, modifying the method of exhaustion by considering not only the inscribed figure but the circumscribed figure as well. The deductive method of exhaustion was not a tool well adapted to the discovery of new results, but Archimedes combined it with infinitesimal considerations toward which Democritus and the Platonic school had groped. The freedom with which he handled these is shown most clearly in the treatise to which we have already referred, the Method.¹³⁵ [ft. 135: For the works of Archimedes in general, see Heiberg, Archimedis opera omnia and T. L. Heath, The Works of Archimedes. For Archimedes' Method, see T. L. Heath, The Method of Archimedes, Recently Discovered by Heiberg; Heiberg and Zeuthen, "Eine neue Schrift des Archimedes"; and Smith, "A Newly Discovered Treatise of Archimedes.'']

This work, addressed to Eratosthenes the geographer, astronomer, and mathematician of Alexandria, was lost and remained largely un- | -known until rediscovered in 1906. (Boyer 48|49)  In it Archimedes disclosed the method which is presumably that which he employed in reaching many of his conclusions in problems involving areas and volumes. Realizing that it is advantageous to have a preliminary notion of the result before carrying through a deductive geometrical demonstration, Archimedes employed for this purpose, in conjunction with his law of the lever, the idea of a surface as made up of lines. For example, he showed that the truth of the proposition that a parabolic segment is 4/3 the triangle having the same base and vertex (the vertex of the segment being taken as the point from which the perpendicular to the base is greatest) is indicated by the following considerations from mechanics.¹³⁶ [Ft 136: See T. L. Heath, The Method of Archimedes, Proposition I, pp. 15-18.]

In the diagram given in Figure 2 [[shown above]], in which V is the vertex of the parabola,

BC is tangent at B,

BD = DP,

and X is any point on AB,

we know from the properties of the parabola that for any position of X we have the ratio

But X" is the center of gravity of XX"',

so that from the law of the lever we see that XX', if brought to P as its midpoint, will balance XX"' in its present position.

This will be true for all positions of X  on AB.

Inasmuch as the triangle ABC consists of the straight lines XX"' in this triangle, and since the parabolic segment AVB is likewise made | up of the lines XX', we can conclude that the triangle ABC in its present position will be in equilibrium at D with the parabolic segment when this is transferred to P as its center of gravity. (Boyer 49|50)

But the center of gravity of ABC is on BD and is 1/3 the distance from D to B, so that the segment AVB is the triangle ABC, or 4/3 the triangle AVB.

This method of Archimedes indicates an anticipation of the use of the concept of the indivisible which was to be made in the fourteenth century and which, when developed again more freely in the seventeenth century, was to lead directly to the procedures of the calculus. The basis of the method is to be found in the assumption of Archimedes that surfaces may be regarded as consisting of lines. We do not know in precisely what sense he intended this to be understood, for he did not speak of the number of elements in each figure as infinite, but said rather that the figure is made up of all the elements in it. That he probably thought of them as mathematical atoms is indicated not only by this manner of expression, but also by the highly suggestive fact that he was led to many new results by a process of balancing, in thought, elements of dissimilar figures, using the principle of the lever precisely as one would in weighing mechanically a collection of thin laminae or material strips.

Using this heuristic method, Archimedes was able to anticipate the integral calculus in achieving a number of remarkable results. He discovered, among other things, the volumes of segments of conoids and cylindrical wedges and the centers of gravity of the semicircle, of parabolic segments, and of segments of a sphere and a paraboloid.¹³⁷ [ft 137: See The Works of Archimedes, Chap. VII, "Anticipations by Archimedes of the Integral Calculus"; also Method.] However, to assert that here "for the first time one can correctly speak of an integration"¹³⁸ is to misinterpret the mathematical process known by this name. [ft 138: Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 154; cf. also p. 155.] The definite integral is defined in mathematics as the limit of an infinite sequence and not as the sum of an infinite number of points, lines, or surfaces.¹³⁹ [ft 139: T. L. Heath (The Method of Archimedes, pp. 8-9) correctly points out that the method here used is not integration; but he gratuitously imputes to Archimedes the concept of the difierential of area.] Infinitesimal considerations, similar to those in the M etlzod, were at a later period to furnish perhaps the strongest incentive to the development of the calculus, but, | as Archimedes realized, they lacked in his time all basis in rigorous thought. (Boyer 50 | 51). This they continued to do until the concepts of variability and limit had been carefully analyzed. For this reason Archimedes considered that this method merely indicated, but did not prove, that the result is correct.¹⁴⁰ [ft. 140:   T. L. Heath, History of Greek Mathematics, II, 29.]

Archimedes employed his heuristic method, therefore, simply as an investigation preliminary to the rigorous demonstration by the method of exhaustion, It was not a generous gesture that led Archimedes to supplement his "mechanical method" by a proof of the results in the rigorous manner of the method of exhaustion; it was, rather, a mathematical necessity. It has been asserted that Archimedes' method "would be quite rigorous enough for us today, although it did not satisfy Archimedes himself."141 [ft 141: T. L. Heath, The Method of Archimedes, p. 10.] Such an assertion is strictly correct only if we ascribe to him our modern doctrines on number, limit, and continuity. This ascription is hardly warranted, inasmuch as Greek geometry was concerned with form rather than with variation. It was, as a result, necessarily unable to frame a satisfactory definition of the infinitesimal, which of necessity was to be regarded as a fixed quantity rather than as an auxiliary variable. Archimedes was probably well aware of the lack of any sound basis for his method and for this reason recast all of his analysis by infinitesimals in the orthodox synthetic form, much as Newton was to do almost nineteen hundred years later after the methods of the calculus had been discovered but still lacked adequate foundation.

[[Demonstration 2]]

The suggestive analysis of the problem of determining the area of a parabolic segment had been given by Archimedes in the Method. However, formal proofs (both mechanical and geometrical) of the proposition were carried out by the method of exhaustion in another treatise, the Quadrature of the Parabola [[see especially propositions 23-24]].¹⁴² [ft 142: See The Works of Archimedes, and T. L. Heath, History of Greek Mathematics, II, 85- 91. A good adaptation of the geometrical proof is given in Smith, History of Mathematics, II, 680-83.] In these proofs Archimedes followed his illustrious predecessors in omitting all reference to the infinite and the infinitesimal. In the geometrical demonstration, for example, he inscribed within the parabolic segment a triangle of area A , having the same base and vertex as the segment. Then within each | of the two smaller segments having the sides of the triangle as bases, he similarly inscribed triangles. Continuing this process, he obtained a series of polygons with an ever-greater number of sides, as illustrated (fig. 3).

He then demonstrated that the area of the nth such polygon was given by the series

where A is the area of the inscribed triangle having the same vertex and base as the segment. The sum to infinity of this series is 4/3A, and it was probably from this fact that Archimedes inferred that the area of the parabolic segment was also 4/3A.¹⁴³ [ft. 143: T. L. Heath, "Greek Geometry with Special Reference to lnfinitesimals."]

However, he did not state the argument in this manner. Instead of finding the limit of the infinite series, he found the sum of n terms and added the remainder, using the equality

As the number of terms becomes greater, the series thus "exhausts" 4/3A only in the Greek sense that the remainder,

can be made as small as desired. This is, of course, exactly the method of proof for the existence of a limit,¹⁴⁴ but Archimedes did not so interpret the argument. [ft.  144 As Miller pointed out in "Some Fundamental Discoveries in Mathematics."] He did not express the idea that there is no remainder in the limit, or that the infinite series is rigorously equal to 4/3A .145 [ft. 146: The Works of Archimedes, p. cxliii.]] Instead, he proved, by the double reductio ad absurdum of the method of exhaustion, that the area of the parabolic segment could be neither greater nor less than4/3A. In order to be able to define 4/3A as the sum of the infinite series, it would have been necessary to develop the general concept of real number. Greek mathematicians did not possess this, so that for them there was always a gap between the real (finite) and the ideal (infinite).

(Boyer, 52)

Above quotation and unmodified images from:

Boyer, Carl. The History of the Calculus and Its Conceptual Development (The Concept of the Calculus). New York: Dover, 1949.