Leibniz "Justification of the Infinitesimal Calculus by that of Ordinary Geometry"

by Corry Shores
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The following will explain Leibniz' "Justification of the Infinitesimal Calculus by that of Ordinary Algebra." Deleuze reads from this letter in his Leibniz courses [Cours Vincennes - 22/04/1980], and he reproduces in The Fold this image from the letter:

[Below is a summary of the letter, taken in incremental steps.]

Gottfried Leibniz

Justification of the Infinitesimal Calculus

by that of Ordinary Algebra

in

Philosophical Papers and Letters

Leibniz constructs the above geometrical diagram:

Straight line AX

and straight line EY

meet at point C

Straight line e forms between E and A, and it is perpendicular to AX.

Likewise, line y forms between X and Y, and it is also perpendicular to AX.

Lines e and y are parallel, because they are perpendicular to the same line.

Line AC is called c,

And all of line AX is called x

We know by Euclid's Proposition 15 that the triangles' angles at C share equal value, because they are opposite or vertical angles.

And we know by Proposition 27 that the other angles must be equal as well

Because all corresponding angles are the same in both triangles, they are similar triangles

This means that the corresponding sides maintain their proportions to each other. We obtain the length of the large triangle's vertical side by subtracting the small triangle's vertical side c from the full line x; so Leibniz describes the proportionality of the triangles' sides by writing

(x - c) / y = c / e

In other words, the large triangle's vertical side is to its horizontal side as the small triangle's vertical side is to its horizontal side.

Now we move line EY closer to point A, always maintaining the angle at C, and hence also at E

Because the angles remain the same, so too will the proportions of their sides, so we can expect the ratio of c to e to remain constant. Leibniz has us assume that this ratio is not 1, for then there would not be a relation of two different values. Thus for this reason, the C angle cannot be 45-degrees; for then the E angle would be as well, which would mean that as an isosceles right triangle, its sides would be the same value. We will see that it is important for this demonstration that there be a difference in value between the terms of the ratio.

Now we imagine that line EY continues toward A and then passes through it.

We see that in the process, points E and C will fall on point A as lines e and c vanish:

We notice it more easily when we magnify the diagram:

And because c has vanished, the vertical side of the larger bottom triangle is no longer defined as x -c, but is rather just x

But we see that as the diagonal line progressed to A, the proportions between the sides of the similar triangles remained the same:

(Animation above is my own, made with Open Office Draw and Unfreeze.)

Looking specifically at the triangle comparisons,

What we see is that the smaller triangle vanished, but as it was vanishing, the larger triangle maintained the proportional relations between c and e. So even though on the one hand we no longer take c and e to have any extensive finite values, we still know that even in their infinitely small size, we can determine their relationship to one another. For, we know that their ratio always remains the same as the larger triangle whose terms retained their extensive finite values. So, we recall that before the contraction, the formula was:

(x - c) / y = c / e

But now it can be rendered:

x / y = c / e

Because now, x - c = x; for c no longer has a finite extensive value to take from x, even though its relation to e still may be represented as

x / y.

So e and c are not nothing, because they preserve the ratio of CX to XY.

Leibniz also says that the ratio of the infinitely small c to e preserves the ratio of the radius and tangent of the angle at C, which would be the same as x and y:

So we know that c and e do not have extensive finite values, but we also know that they are not nothing. For, if they both equaled zero, then x / y would equal 0 / 0, which equals 1 [0r 1/1]. Hence in that case, x and y would have the same value, and thus x would equal y. However, this is absurd, because it could only be so if C's angle were 45-degrees. But we began by presuming that it was not.

So c and e are not taken to be zero, except in their relation to x and y. But when they are in a ratio to each other, c and e have an algebraic relation to one another.

And so they are treated as infinitesimals, exactly as are the elements which our differential calculus recognizes in the ordinates of curves for momentary increments and decrements.

(545d)

Although some reject the notion of the infinitesimal, it proves advantageous in solving certain algebraic problems that otherwise would be insoluble. As well in physics this principle may take the form of Leibniz's Law of Continuity, by which one may regard

equality as a particular case of inequality, rest as a special case of motion, parallelism as a case of convergence, etc., assuming not that the difference of magnitudes which become equal is already zero but that it is in the act of vanishing; and similarly in the case of motion, not that it is already zero in an absolute sense but that it is on the point of becoming zero.

(546b)

This sort of method was used by Archimedes; and anyone who criticized his use of infinitely small values would yet be unable to show how we may designate some magnitude for them.

Thus,

rest, equality, and the circle terminate the motions, the inequalities, and the regular polygons which arrive at them by a continuous change and vanish in them. And although these terminations are excluded, that is, are not included in any rigorous sense in the variables which they limit, they nevertheless have the same properties as if they were included in the series, in accordance with the language of infinites and infinitesimals, which takes the circle, for example, as a regular polygon with an infinite number of sides. Otherwise the law of continuity would be violated, namely, that since we can move from polygons to a circle by a continuous change and without making a leap, it is also necessary not to make a leap in passing from the properties of polygons to those of a circle.

(546c.d)

Leibniz, Gottfried. "Justification of the Infinitesimal Calculus by that of Ordinary Algebra." Philosophical Papers and Letters. Ed. & Transl. Leroy E. Loemker. Dordrecht: D. Reidel Publishing Company, 1956.