The following are statements attributed to Leonardo in his written works [1]. Each paragraph is a different statement from a potentially different original source.

1. But since we know that painting embraces and contains within itself all things produced by nature or whatever results from man’s passing actions – and ultimately everything that can be taken in by the eyes (…) he seems to me to be a pitiful master who can only do one thing well.

2. The good painter has to paint two principal things, that is to say, man and the intention of his mind. The first is easy and the second difficult, because the latter has to be represented through gestures and movements of the limbs.

3. I shall not refrain from including among these precepts a new aid to contemplation, which, although seemingly trivial and almost ridiculous, is none the less of great utility in arousing the mind to various inventions. And this is, if you look at any walls soiled with a variety of stains, or stones with variegated patterns (…), you will therein be able to see a resemblance to various landscapes graced with mountains, rivers, rocks, trees, plains, great valleys and hills in many combinations. Or again you will be able to see various battles and figures darting about, strange-looking faces and costumes, and an endless number of things that you can distil into finely rendered forms. And what happens with regard to such walls and variegated stones is just as with the sound of bells, in whose peal you will find any name or word you care to imagine.

4. I say that when you are painting you ought to have by you a flat mirror in which you should often look at your work. The work will appear to you in reverse and will seem to be by the hand of another master and thereby you will better judge its faults.

5. How to learn well by heart: When you wish to be able to make use of something committed to memory adopt this method, which is that when you have drawn the same thing so many times that it seems you have it by heart try to do it without the exemplar. Have your exemplar traced on to a thin flat plane of glass. Place this on top of the drawing you have done without the exemplar, and note carefully where the tracing does not match up with your drawing, and in those places where you have made a mistake, resolve not to repeat the error. In fact, go back to the exemplar and draw over and over it the erroneous part till you have it firmly in your memory.

6. I say that in narrative paintings you should closely intermingle direct opposites, because they offer a great contrast to each other, and the more so the more they are adjacent. Thus, have the ugly one next to the beautiful, the large next to the small, the old next to the young, the strong next to the weak. In this way there is as much variety, as closely juxtaposed as possible.

7. It previously happened to me that I made a picture representing a holy subject, which was bought by someone who loved it and who wished to remove the attributes of its divinity in order that he might kiss it without guilt. But finally his conscience overcame his sighs and lust, and he was forced to banish it from his house.

8. And if you should have a love for such things you might be prevented by loathing, and if that did not prevent you, you might be deterred by the fear of living in the night hours in the company of those corpses, quartered and flayed and horrible to see. And if this did not prevent you, perhaps you might not be able to draw so well.

Do you know what you will achieve if you practice drawing with a pen? It will enable you, trained and skilful, to draw a great deal in your head.
Exert yourself and take delight in copying always the best things – crafted by the hand of great masters – that you can find.

Cenninno Cenninni, Libro d’Arte, 1400

In the doctor’s schools of anatomy he dissected the corpses of criminals, undismayed by the brutal and repulsive nature of this study and only eager to learn how to portray in his painting the various limbs and muscles, their bending and stretching, in accordance with the laws of nature.

Paolo Giovio, 1525

I would not like to neglect to repeat the words I heard King François I say about him: (…) he did not believe there could be anyone else on earth who knew as much as Leonardo, not just about sculpture, painting and architecture, but also insofar as he was a great philosopher.

Benvenuto Cellini, 1562

Altogether, his genius was so wonderfully inspired by the grace of God, his powers of expression were so powerfully fed by a willing memory and intellect, and his writing conveyed his ideas so precisely, that his arguments and reasonings confounded the most formidable critics.

Giorgio Vasari, 1568

During his apprenticeship, Leonardo sometimes made clay models, draping the figures with rags dipped in plaster, and then drawing them painstakingly on fine Rheims cloth or prepared linen. These drawings were done in black and white with the point of the brush, and the results were marvellous, as one can see from the examples I have in my book of drawings.

Giorgio Vasari, 1568

But before we go any further, we must say a little more about Leonardo’s personality and talents. The many gifts that Nature bestowed upon him concentrated themselves primarily in his eye. Hence, although capable of all things, he appeared great above all as a painter. He did not rely simply upon the inner impulses of his innate, inestimable talent; he permitted no arbitrary, random stroke of the brush; everything had to be deliberate and considered. From the pure proportions to which he devoted so much research, to the strangest monsters that he compiled out of contradictory figures, everything had to be both natural and rational.

Goethe, 1787

It requires a quite different strenght and agility to maintain one’s position within a system that is never fixed and where ideas are free and still evolving, than in a dogmatic world. Leonardo da Vinci stands higher than Michaelangelo, Michaelangelo higher than Rafael.

Nietzsche, 1885

Painting, for Leonardo, is an operation which calls for every sphere of knowledge and almost every technique. Geometry, dynamics, geology, physiology. Representing a battle requires a study of whirlpools and swirls of dust; he will only portray them having observed them with his own eyes, so that his attempt will be well researched and informed by an understanding of their laws.

Paul Valéry, 1895

Art and scientific genius came together in Leonardo’s spirit.

Thomas Mann, 1936

Tolerance, it is true, demands that we respect differences of belief. But as already recognized by Leonardo da Vinci, to the degree that the truth becomes better known, so general consensus will come to replace individual opinions.

Thomas Mann, 1945

______________

[1] Source: Leonardo da Vinci, The Complete Drawings by Johannes Nathan and Frank Zöllner, Taschen, Bibliotheca Universalis, 2023

Proper, scientific machine Learning.
Proper, scientific machine Reasoning.
The engineering Middle Ages ends today.

Arithmetic and Geometry offer the supreme certainty.

Leonardo da Vinci

The presentation, without the demonstration, can be downloaded here.
You can watch the video on Rumble here.

This model is based on the proper, scientific machine model of Learning which is described here.

February 2026 Update:

This model became the basis for a proper, scientific machine model of Cognition which is described here.

____________
Header image: citological drawing by Ramon y Cajal.

I have built a pyramid from scratch.
Starting from the height of the blocks (0.5 cm) and the desired width of the base (7.5 cm) and height (5.5 cm) of the pyramid, I computed the width offset between layers and with it the 2D dimensions of the blocks in each layer.
I used 48 blocks with different dimensions stacked in 9 layers.

The calculations are as follows:
h’ divides h in as many parts as x’ divides x.
Therefore, h = k h’ and x = k x’ with k natural and equal to the number of blocks in height.
Therefore, x’ = x / k = x h’ / h.
In a pyramid, the base side will be L = 2x, so x’ = L h’ / (2 h).
The width of the n-th layer will be L(n) = L – 2 x’ n.

The dimensions of the block in each layer depend on the architecture which I chose to be such as to preserve the maximum number of blocks with their original standard dimension which is 2.5 x 2.5 cm. This architecture gives 3 types of blocks which are the standard one, the one with one dimension dependant on the layer and another one with both dimensions dependant on the layer.

I made the blocks with air dry clay and I used white cement and ocher yellow powder for the external cover.