Divided line

The Divided Line

Analysis of the "strange" property

Geometric construction and proof of the two-segment equality

by Tudor B. Munteanu

Analysis

In The Republic Book VI, Plato explains the division of the line corresponding to the spheres of reality:

"... Now take a line which has been cut into two unequal parts, and divide each of them again in the same proportion, and suppose the two main divisions to answer, one to the visible and the other to the intelligible, and then compare the subdivisions in respect of their clearness and want of clearness" (Republic, 509d-e, translated by Benjamin Jowett)

The resulting sections are the visible, consisting of two subdivisions for images (shadows) and things ("figures", or objects) - and the intellectual, with its subdivisions for concepts used in hypotheses, and forms ("ideas" - eide). The corresponding faculties are: imagination, belief (conviction) - understanding and reason (nous). As Plato writes, "the several faculties have clearness in the same degree that their objects have truth."

The Divided line, image taken from http://www.friesian.com/plato.htm

Some commentators have found it strange that the two segments corresponding to objects or conviction, on one hand, and concepts or understanding, on the other - have the same length. A few even suggested that the lengths or any proportion shouldn't matter, i.e. all segments could be equal; others insist that the proportion should be in "mean and extreme ratio", i.e. "the golden mean". Even a cursory reading of "The Republic" conveys the importance of constructing the segments of the "Divided Line" proportionally, according to the analogy: the relationships between the different sections correspond to the proportional relationships and significance of the segments. I have found no reason for a specific proportion, such as the "golden mean". An exhaustive analysis of this possibility can be found in Dr. Yuri Balashov's paper, (pdf) "Should Plato's Line Be Divided in the Mean and Extreme Ratio? published in Ancient Philosophy 14 (1994): 283-95

The equality is not discussed explicitely in the Republic, but it appears in another dialogue, the Phaedo, where the hypothetical method is presented in the analogy of the eclipse:

"But perhaps my analogy is in a way not quite accurate, for I do not at all agree that someone who examines beings in propositions is examining them in images any more than someone who examines them in things [erga]." (Phaedo, 99e6–100a3, translated by Francisco J. Gonzalez)

Socrates' hesitation was explained in "The Transcendent Way of Knowledge", my review of Francisco J. Gonzalez' book, "Dialectic and Dialogue: Plato's Practice of Philosophical Inquiry".

The section corresponding to understanding (dianoia) is superior on account of its increasing clarity; concepts are predicated of objects, whose relationships are understood and can be expressed only through other concepts, etc. that must rely at some point on images or ideas. Relations among the subsections of the Divided Line follow the proportion(s) in a strict manner: ideas are to concepts as objects are to images, or going accross the two realms - ideas are to objects as concepts are to images. One cannot relate objects and concepts directly, only through images or through ideas - and this is the significance of the equality: what is predicated must refer to a content, either through an image that contains sensations, or through a higher hypothesis that reflects a relation of participation, which can ultimately be justified only by a fundamental, intellectual intuition (noesis) or by recollection. Therefore, after Plato, only the "overcoming" of hypotheses through the eide can justify knowledge proper.

Geometric construction

The following geometric construction and demonstration features David E. Joyce's Geometry Applet, from his beautiful online edition of Euclid's Elements (links). The drawing, demonstrations and all mistakes, omissions or innacuracies are mine.

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Draw line AB, and to divide choose point C on AB so AC<CB. Point C defines the proportion between the segments AC and CB of the Divided Line. Note: point C is "movable" i.e. it can be moved freely by dragging within the applet image at right. The whole figure can be moved within its boundaries by dragging any fixed, i.e. non-movable point.

These sections, AC and CB will be divided in two segments each, in the same proportion, as follows (applying two constructions similar to Book VI, proposition 11; all such references are to Euclid's Elements)

Draw line CK, anywhere from C and choose points O and D so CO=AC and OD=CB (for a rigorous construction, follow Book I, proposition 2)

Therefore, O divides CD in the same proportion as C divides AB. Note: point K, just like C is movable: it can be moved freely by dragging. Moving both these points contributes to a better understanding of the construction.

Also, Draw AE=CD parallel to CD so AEDC is a parallelogram (Book I, proposition 11). Similarily, choose point P on AE so ACOP is a parallelogram. AP=CO=AC and PE=OD=CB, as per our construction.

Draw ON parallel to DB, so N belongs to line CB. triangles CON and CDB are similar, N divides segment CB in the same proportion as C divides AB and O divides CD.

Draw EC that intersects PO at point Q. Draw PM Parallel to EC, so M belongs to AC. Because triangles APM and AEC are similar, M divides segment AC in the same proportion as C divides AB and P divides AE.

Geometric proof of the "strange property".

Note: to see the figure, shift down the drawing by dragging a fixed point, e.g. point D.

It is fairly obvious from the figure that two subsections, MC and CN are equal, but this conjecture is based on perception of images, and must be explored in understanding through rigorous hypotheses based on more fundamental hypotheses, "axioms" relying on pure intuitions, in Kant's sense - as in Euclid's Elements (mathematical objects have the same epistemological value as any perception: belief, in the sense of trust).

MC=PQ, because MCQP is a parallelogram. It seems natural to draw PC and QN, and show that PQNC is also a parallelogram, i.e. PC is parallel to QN. If that is true, then PQ=CN so MC and CN would also be equal. It follows:

PC is parallel to EB, because triangle APC is similar to triangle AEB (PC parallel to EB) due to the construction ( AP/PE = AC/CB)

but CQ/QE = CO/OD because triangle CQO is similar to triangle CED (PO, i.e. QO parallel to ED) also due to the construction.
also CO/OD = CN/NB because triangle CON is similar to triangle CDB (ON parallel to DB), also from the construction.

it follows from the above equalities that CQ/QE = CN/NB, so triangle CQN is also similar to triangle CEB, therefore QN is also parallel to EB.

since PC and QN are parallel to EB, PC is parallel to QN, Q.E.D. Therefore, MC=CN because PCNQ and PQCM are both parallelograms with a common side, PQ.

Arithmetic proof of the equality.

The arithmetic proof does not rely on any images. Although usually we look for something intuited (in a vague sense), given the proportions one could find the equality just by playing with the arithmetic relations given in our proportion, in a manner similar to formal logic - but nobody would know the significance of our discovery if we didn't provide some meaning, or content e.g. an illustration or any interpretation. The same applies to the equations and to the axioms of mathematics.

The constructed proportion was
AM/MC = CN/NB = AC/CB [= (AM+MC)/(CN+NB)] (1)
[AC/CB = (AM+MC)/(CN+NB) because AC=AM + MC and CB = CN + NB]

a derived proportion can be obtained by adding terms in the denominator:

AM/(AM+MC) = CN/(CN+NB) therefore -
denominator over denominator: (AM+MC)/(CN+NB) = AM/CN

also from the initial proportion (1), (AM+MC)/(CN+NB) = AM/MC

from the above two equalities, it follows that AM/MC = AM/CN, therefore MC = CN