Introduction

We must seek the causes of the harmonic proportions in the divisions of a circle into equal aliquot parts, which are made geometrically and knowably, that is, from the constructible regular plane shapes.

I thus considered that to start with it should be intimated that the features which distinguish geometrical objects to the mind are today, as far as is apparent from published books, totally unknown.

In fact, among the ancients, only Euclid and his commentator Proclus could say that knew exactly these specific 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 the 10th book 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
• the unlimited as the material of geometrical objects

This is because shape and proportion are properties of:

• quantities, and
• shape of individual quantities and proportion of 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 things which are shaped can also be grasped by the mind through definitions and bonds of constructions
• infinite things cannot be known

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 mental essence of them. 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 the tenth Book of Euclid.

He who had written a commentary on Euclid was angrily repudiated by Petrus Ramus who condemned Euclid’s Book 10:

But my goodness, Ramus, if you had not believed that this book was too hard to understand, you would never have slandered it with the accusation of such obscurity?

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

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

When the superior mind has struggled to that point, then at last, seeing that it has reached the light of truth, it is exultantly flooded with incredible pleasure, and in that, as it were, watch tower, it perceives with great precision the whole world and all the distinguishing features of its parts.

But you who :

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

You make “prodigious sophisms” as your comments of:

• “Euclid incontinently abusing his leisure”
• “these subtleties have no place in geometry”

Let your part be to carp at what you do not understand.

I am a hunter for the causes of things. Only Book 10 of Euclid opened the paths to them.