Proclus Diadochus

Mathematics contributes very important things to the study of nature. It reveals the orderly nature of the reasoning, in accordance with which the whole has been constructed, and showing that the primary elements took on the appropriate forms and are connected together with symmetry and regularity.

Proclus Diadochus

Commentary on Euclid Book 1

We must seek the causes of the harmonic proportions in the divisions of a circle into equal aliquot parts.

• These are made geometrically from the constructible regular plane shapes.

It is totally unknown how the mind distinguishes geometrical objects.

In fact, among the ancients, only Euclid and his commentator Proclus knew exactly these distinguishing features of geometrical objects.

Every single part of the subject of geometry creates certain mental attitudes.

These attitudes can be explained by dividing problems into planes, solids, and lines, as done by Pappus of Alexandria and his followers.

However he is both:

• brief in words
• focused on practice

He never mentions his theory. We need to know his theory otherwise we never will be able to take in the harmonic ratios.

Proclus Diadochus published 4 books on the first book of Euclid. He played the part of a theoretical philosopher dealing with a mathematical subject.

His commentaries on book 10 of Euclid has relieved me totally from this toil of explaining the distinguishing features of geometrical objects.

Geometrical objects are entities of the mind which are different from each other, just as living entities are different.

Euclid established the basic principles of the whole essence of mathematics to know such differences such as:

• the end versus the endless
• the limit versus the unlimited

He recognized that:

• the limit as the form [shape]
• the unlimited [line] as the material of geometrical objects

Shape and proportion in general are properties of quantities.

The shape of individual quantities and proportion are properties of those quantities combined.

Shape is demarcated by limits.

Examples of such limits are:

• the points that limit a straight line
• the lines that limit a plane surface
• surfaces that limit a 3D solid

Therefore:

• finite shapes can also be grasped by the mind through definitions and bonds of constructions
• infinite things cannot be known [defined]

Shapes are:

• an archetype prior to becoming produced
• in the divine mind prior to being in creatures
• different in respect of their subject, but the same in the form of their essence.

Therefore, in quantities, shape is a kind of their mental essence.

This essence is made clearer through proportions.

Shape is demarcated by several limits.

Shapes gain proportion when they are many.

However, this proportion, without the action of the mind, is something which cannot be understood in any way.

Hence by the same reasoning, one who gives limits to quantities as their essential basis supposes that quantities which have shapes have an intellectual essence.

But there is no need for arguments, as explained by Proclus’ book.

The intellectual distinguishing features of geometrical objects were properly known to him.

• He is completely an abstruse Platonist.

However, our age has had no room to penetrate such hidden mysteries.

Proclus’ book was read by Petrus Ramus who despised and rejected it equally with Book 10 of Euclid.

The subject material of Book 10 is so obscure. It thoroughly understands and investigates its goals such as the kinds and distinguishing features of objects. I have never read or heard anything so confused. The superstition of the Pythagoreans seems to have invaded this cave.

Ramus

But my goodness, Ramus, if it was too hard to understand why would you slander it as obscure?

Until you grasp the writer’s intention, you need:

• harder work
• tranquillity
• concentration
• mental exertion, above all

After the mind has struggled to that point and has reached the light of truth, it will perceive with precision the whole world and all the distinguishing features of its parts.

But you:

• act as the patron of ignorance with the common herd of men, and
• snatch at profit from everything, divine or human