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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.
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