Nathaniel Perrin

A translation of Bernhard Riemann's inaugural lecture

Riemann: 'On the Hypotheses which Lie at the Root of Geometry'



### Preface

Why Uexküll?

Many thinkers have unsung fathers. Forerunners who begot the ideas that their successors were celebrated for. (Who remembers Brunschvicg, Polanyi, or Henry of Ghent?) In this case, both Heidegger and Deleuze owe some of their thought to the good Baron Uexküll. 

Philosophers are often so distant from any concrete form of experience or work, their prestige garnished from verbal posturing and polemics, in a way sometimes no better than middle management. Not so with Uexküll: he was a biologist. A son of the Baltic, a German aristocrat living in what is present-day Estonia.

Uexküll's writings on the consciousness of animals, among other ideas, inspired developments in fields as wide ranging as phenomenology, semiotics, cybernetics, as well as in the biological sciences.

If this should not make him intriguing enough, Uexküll was highly critical of Darwinism from a theoretical standpoint. 

Grasping the bankruptcy of a scientific theory derived from liberal capitalism, Uexküll hoped to develop a new biological outlook as an alternative to Darwin's shallow theory. That biology is not at its root a sequence of individuals eliminated, or not. But instead a biology focused on the nexus of relationships between animals and the world, starting from the viewpoint that different species belong together and mutually determine each other.

Unfortunately, very little of Uexküll's work has been translated into English. The only full-length book I have been able to find is *A Foray into the Worlds of Animals and Humans with a Theory of Meaning* which combines *Streifzüge durch die Umwelten von Tieren und Menschen* with *Bedeutungslehre*.

Here I will present a translation of the introduction to one of his earliest publications, *Umwelt und Innenwelt der Tiere* (1909) which can be found [here for free](https://www.biodiversitylibrary.org/item/15902#page/5/mode/1up). I translated this above as "The External and Internal Worlds of Animals" to reflect the dual natures of the world outside *Um-welt* and the world inside *Innen-welt*, but it should be noted that *Umwelt* is the German word for the "environment", that is the world all around us. One of those colorful forms of German wordplay that cannot be translated succinctly.


I will likely translate more of the book. If you are interested in fetching a draft copy, feel free to [reach out](https://www.ngperrin.com/contact).

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### Introduction

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These days, the word "science" is the subject of ridiculous fetishism. Therefore, it is probably appropriate to point out that science is nothing but the sum of the opinions of researchers alive today. As far as we record the opinions of older researchers, they also live on in science. As soon as an opinion is discarded or forgotten, it is dead to science.


Little by little, all opinions are forgotten, discarded, or modified. Therefore one can answer the question, "What is a scientific truth?" without exaggeration with: "The misconception of the day."

The question, whether there is progress in science, is therefore not quite as simple to answer as is generally assumed. We hope to progress from cruder to more refined errors, but whether we are really on the right track for biology is highly doubtful. The study of the living offers such an immense abundance of facts to the unbiased observer at every step that the mere registering of these facts would make any science impossible. Only the opinion of the researcher, which forcibly separates the studied into the essential and the unessential, gives rise to science. The prevailing opinion ruthlessly decides on what should be considered "essential". If it is overturned, thousands of diligent, painstaking, and excellent studies will fall into oblivion along with it as "unessential".

In biology, we are still under the fresh impression that the fall of Darwinism has produced in all of us. The achievements of half a century of restless work today seem insignificant to us.

No wonder that biologists now strive to give their work a firmer foundation than the doctrine of the perfection of living things had. 

The success of these efforts is not very encouraging. No agreement has yet been reached on the foundations upon which animal biology is to rise as a proud scientific house. And yet this agreement decides the fate of biology. If the question of the foundations remains undecided or subject to fashionable trends, there will be no progress, and everything that has been worked out with the greatest effort of mind by one generation will be discarded by the next.

Only when all hands work according to a shared plan to build a house on a firm foundation that something prosperous and lasting can come into being.

It is instructive and perhaps useful to be clear about the causes that have prevented agreement in modern animal biology so far.

Modern animal biology owes its existence to the introduction of physiological experimentation into the study of lower animals. The expectations that were attached to the expansion of the field of research from the physiological side were not fulfilled. One looked for a solution for the questions of the physiology of higher animals and found new problems instead. The dissolution of living phenomena into chemical and physical processes did not advance it one step. Thus experimental biology discredited itself of more rigorous observance in the eyes of the physiologists.

For all those researchers, however, who saw the essential content of biology in the life process itself and not in its reduction to chemistry, physics, and mathematics, the enormous wealth of experimentally solvable problems had to be a special incentive to turn to the lower animals. In a few years the abundance of the edited material has become so large that today the ordering of the material appears as the much more urgent task, vis-à-vis the always progressing new research. Building material is available in abundance to start the construction of science. One must only agree upon the construction plan.

The most natural thing to do would be to continue working with the old, long-standing construction plans. In the good old days, when anatomy and physiology still formed a unified biology, each animal was regarded as a functional unit. The anatomic structure and its physiological performances were studied simultaneously and considered to belong together.

It does not occur to anyone to introduce a division of labor into technology, and to train two classes of engineers, one for the study of structure, the other for the study of energy conversion in machines.

Technology, like engineering, would soon be ruined by this division. Biology, too, would have perished long ago from division into anatomy and physiology, if medicine with its practical requirements had not peremptorily demanded the union of the two sciences, at least for man. The latest work also owe its deep biological significance to this joining of the sciences. One need only remember the life work of Pavlov, or the great success of English physiologists like Langley and Sherrington.

Wherever physiology and anatomy proceeded separately, it did not turn out for their good. Comparative anatomy, which more and more neglected the merit of the organs, finally came to consider the structure of the living beings as a mere "formal unity". "Homology" became the basis of a completely new doctrine of the relationships of body types, while analogy was despised, and thus dead spatial relationships took the place of the living interaction of organs. Only in recent times does experimental embryology lead anatomical science back to the sources of the deepest problems of life. 

Likewise, general physiology increasingly lost the understanding that every living being is a "functional unity". In place of striving for knowledge of the structural plan of each living being, which can be deduced from anatomy and physiology alone, the one-sided study of the most isolated partial functions ever possible took place, in order to be able to handle these as purely physical or chemical problems.

This was the doom of the biology of higher animals. Quite strange was the fate of the biology of lower animals. Here anatomy and physiology did not go separate ways, but physiology was completely suppressed temporarily. This happened by Darwinism. Darwinism (not Darwin himself) regarded the capacities of anatomical structure as "unessential" in contrast to a problem: how the structure of higher animals had developed from that of lower ones.

One saw in the series of evolving animals the proof for a increasing perfection step-by-step from the simplest to the most multifaceted structure. Only one thing was forgotten, that the perfection of the structure cannot be derived from its multifaceted nature. No one will claim that an ironclad is more perfect than the modern rowboats of the international rowing clubs. Also, an ironclad would play a pitiable role in a rowing regatta. Likewise, a horse would fill the role of an earthworm only very imperfectly.

The question of a higher or lower degree of perfection of living beings can only be asked if one checks every blueprint with its execution and sees in which case the execution is most successful. In this question, there is no doubt the lower animals will win the prize, since they belong to the oldest creatures, because this rule seems to apply: the older the family, the better the performance.

One tries to discuss the problem of perfection by comparing the needs of the organisms with their blueprint, and thus asks, to what extent this blueprint corresponds to its needs. This has also been the question of Darwinism. Only from this does the assertion hold meaning that higher animals are more perfect.

For if the needs of man are regarded as the measure against which all the blueprints of animals are to be measured, then naturally the highest animals are the most perfect. But this is too evident an error to waste a word on it. After all, we have no other means at hand for the investigation of the needs of an animal other than its blueprint. That alone gives us information about the active and passive role which the animal is appointed to play in its environment. Therefore the whole question is meaningless.

But even the assertion that the varying individuals of a species are more or less well adapted to their environment is completely taken out of thin air. Every varying individual is different according to its modified blueprint but somehow also equally perfectly adapted to its environment. Because the blueprint on its own establishes the environment of the animal with wide limits.

This realization, which I intend to prove step by step, can alone be regarded as the permanent basis of biology. Only through this can we gain the correct understanding of how living beings order and control the chaos of the inorganic world. Every animal in its own place and its own way. Out of the immense diversity of the inorganic world, every animals chooses just that which suits it (i.e. it creates its own needs according to its design).

Only at a superficial glance does it appear as if all sea animals live in a shared homogeneous world. Closer study teaches us that each of these thousandfold varying life forms possesses an environment unique to it, which mutually conditions itself with the blueprint of the animal.

It is not surprising that the environment of an animal also includes other living beings. Thus this mutual conditioning also takes place between the animals themselves and results in a strange phenomenon that the predator fits to the prey just as well as the prey is fitted to the predator. Thus no only is the parasite adapted to the host, but also the host to the parasite.

The attempt to explain this mutual belonging togetherness of neighboring animals through gradual adaptation has failed miserably. It has also diverted interest from the nearest task, which is to first determine the environment of each animal.

This task is not as simple as the novice might believe. It is not difficult to observe any animal in its environment. But this in no way resolves the problem. The experimenter must try to determine which parts of this environment affect the animal and in what way.

Our anthropocentric way of looking at things must step back more and more, and the point of view of the animal must become the sole decisive one.

With that, everything which is considered natural to us disappears: the whole nature, the earth, the sky, the stars, yes all objects which surround us, and only those influences which exert an influence on the animal according to its blueprint remain as world-factors. Their number, their association is determined by the blueprint. If this connection of the blueprint with external factors is carefully investigated, then a new world, completely different from ours, is formed around each animal, as its environment.

It is just as objective as the factors of the environment are, the effects caused by them must also be acknowledged and regulated by the blueprint. Together they form the inner world of animals.

This inner world is the unadulterated fruit of objective research and is not to be clouded by psychological speculation. One may perhaps, to make the impression of such an inner world lifelike, raise the question, what would our soul do with such a limited inner world? But to paint and to spruce up this inner world with mental qualities (which we can neither prove nor deny) is no activity for serious researchers.

The blueprint stands above the inner world and the environment, dominating everything. In my opinion, the research of the blueprint alone can provide a healthy and secure basis for biology. It also brings anatomy and physiology together again for more fruitful interaction.

If the development of the blueprint for each animal species is placed at the center of research, then every newly discovered fact finds its natural place, where it receives meaning and significance.

The contents of the present book shall serve the purpose of demonstrating the importance of the blueprint as vividly as possible and to show by individual examples how environment and inner world are interrelated through the blueprint. A textbook of special biology is not offered here, but only the way is shown, by which one could arrive at one.

In the selection of present examples, I was determined above all by the desire to give pictures as systematically as possible. Of course, there are gaps everywhere, not only in the physiological but also in the anatomical material. On the other hand, since I could only use such anatomical material that was physiologically alive, the great mass of anatomical and zoological knowledge had to be omitted. Likewise all physiological results had to be neglected, which offer only physical or chemical interest. But even those structures whose capacities are well studied had to be disregarded if their complication made too great demands on the reader's imagination.

Finally I have limited myself to the invertebrates, because I am at home there myself, leaving the higher animals to the more qualified. From the invertebrates, bees and ants remained unconsidered, because detailed textbooks about them are already available.

I could now proceed to the content of the book, because the point of view, from which it is to be considered, is sufficiently stated. But it is unnecessary to go into those opinions that strive to give biology a different basis.

What can arise in the way just outlined is a special biology of all animal species. Such a biology would be very one-sided if it did without the aid of comparison. All animals perform their animalistic capacities with the help of tissues which remain very similar through the whole set of animals. Muscle tissue and nerve tissue show analogous capacities everywhere, no matter how differently they are combined in organs. This is of great importance for special biology, because the generally valid properties of muscle and nerves can be presupposed as valid also in those animals whose body constitution does not permit a physiological analysis down to the individual tissues. Therefore, the comparative physiology of the tissues will always remain a very necessary part of special biology, and there is nothing to be said against it, if the comparative tissue science precedes the discussion of the individual animals. I have refrained from doing this because I wanted to show in which animal species we can most easily reach more general conclusions for general tissue science.

The biology is completely different, if one makes the comparison the basis of the entire study. This has been done by Loeb, and in an extraordinarily original and interesting way.

The great majority of animal movements proceed as follows: An external stimulus acts on a receptive organ, which gives an excitation to the nervous system. Directed by the nervous system, the excitation finally reaches the muscle, which then contracts. This process is called a reflex. Loeb found that a large number of animals, when exposed to very elementary stimuli, such as light, gravity or simple chemical substances, always respond with an ordered movement, by which they either turn toward or away from the source of the stimulus. He saw in this an elementary process, which he called tropism, and depending on the direction taken by the movement, he spoke of positive or negative tropism.

Loeb himself admitted the possibility that many tropisms are not yet sufficiently analyzed reflexes. But certain tropisms, e.g. the phototropism, which occurs on one-sided exposure, he wants to be regarded as an elementary phenomenon to be equated with the physical phenomena. The light rays should be able to turn the animal body during their passage through it like a magnet turns iron filings. Animals which react to light in this way are called photopathic.

But there is no doubt that in many cases the light simply causes a reflex on the illuminated side of the animal, which must result in a unidirectional movement, since there is no reflex on the shaded side. The animals that react in this way against the light are called phototactic.

Photopathic phototropism is a physical process, whereas phototactic is a reflex.

Now Ms. Lee has been able to prove on unicellular animals that the photopathic explanation of their movements can very well be replaced by a phototactic one.

Recently Radl has tried to prove that light has the same directional effect on insects as gravity has on a floating body. On the other hand, G. Bohn has found that the undeniably directional effect of illuminated objects on snails and crabs depends on the physiological state of the animals.

One sees from this, how uncertain the interpretation of these processes is.

It seems tempting to attribute all movements of the animals to tropisms, because this relieves us of the task of treating the apparently simple processes as performances of a structure which is difficult to determine. But a certain basis is gained only by the study of the structure and the blueprint.

Already this view seems to gain more and more ground. But only some of the researchers turns to the study of the blueprint. Another part follows a new doctrine, which rejects the study of the blueprint and wants to regard the animals free from any analogy with the machines.

It is undeniable that determining the blueprint of the animals has only then a sense, if the structure of the animals is to be equated with the structure of the machines.

With this we approach the basis of all biology, which cannot be decided by speculation, but only by observation of the living substance, on which all living beings are built, while the machines consist of dead material-- we approach the protoplasm problem.

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Translation completed with assistance from [DeepL](www.deepl.com).

The introduction from Umwelt und Innenwelt der Tiere, in English for the first time

Uexküll translation: The External and Internal Worlds of Animals

Meeting notes of the 1922 encounter translated from the French, perhaps for the first time

The infamous Bergson-Einstein debate, translated

Figure 92. A diagram of the geometrical proof offered in the Greek fragment of Heron's *Catoptrica*. Taken from the critical edition *Herons von Alexandria Mechanik und Katoptrik* (Teuber, 1900), pg. 371. Schmidt himself notes his printed diagram is taken from Aldina (1551).

This is likely a reference to the [imprint belonging to the Venetian humanist Aldo Manuzio](https://it.wikipedia.org/wiki/Aldina), as he published many Greek and Latin editions. However, it is not clear in which volume one would find this diagram, particularly because the imprint only published between 1495 and Manuzio's death in 1515, which is a few years short of 1551 (unless Schmidt is guilty of a numerical typo).

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**As is my habit to release posts early and return to them often, the reader should be warned this one is still a "work in progress".**


## Introduction

An introduction. Heron of Alexandria was perhaps one of the most ingenious inventors of Western antiquity, if not all history. Living in the first century, his catalogue of inventions includes a prototype windmill, mechanical theater technology, a water fountain, syringes, and the first coin-based vending machine. More than this, Heron had detailed design notes for the first steam engine, a technology which would not resurface for another 1500 years. It is remarkable a man living in the same century as Augustus and Jesus would designed such technologies.

Heron is a remarkable demonstration of the fact that historical anachronism can be found in real life as well.

Many of Heron's works do not survive to this day, as is the case for much literature from antiquity. Some works such as the *Mechanica* come to us through Arabic translation. Not all of his extant works have been translated into English either. It was only in 2019, that [a PhD thesis translated](https://theses.gla.ac.uk/76774/) his *Automata* into English.

My interest in him comes from his original contributions to classical optics and thus the history of science.

His *Catoptrica*, one of the most significant works in the history of optics, has not yet made it into English. I will attempt to remedy this by starting with the single extant Greek fragment, and then proceeding through a translation of the entire Latin work into English in a subsequent post.

Comments and feedback are always appreciated, so long as they do not steer too deeply into the pedantic.

Let us begin.







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## The Text

The text is taken W. Schmidt's critical edition *Herons von Alexandria Mechanik und Katoptrik* published by Teuber in 1900, pages 368-373. I referred to the German translation in a few spots but will offer more analysis of it and textual variants in the forthcoming Commentary section.

ἐπειδὴ γὰρ τοῦτο ὡμολογημένον ἐστὶ παρὰ πᾶσιν, ὃτι οὐδὲν μάτην ἐργάζεται ἡ φύσις οὐδὲ ματαιοπονεῖ, ἐὰν μὴ δώσωμεν πρὸς ἲσας γωνὶας γὶνεσθαι τὴν ἀνάκλασιν, πρὸς ἀνὶσους ματαιοπονεῖ ἡ φύσις, καὶ ἀντὶ τοῦ διὰ βραχείας περιόδου φθάσαι τὸ ὁρώμενον τὴν ὂψιν, διὰ μακρᾶς περιόδου τοῦτο φανήσεται τὴν ὂψιν, διὰ μακρᾶς περιόδου τοῦτο φανήσεται καταλαμβάνουσα. εὑρεθήσονται γὰρ αἱ τὰς ἀνίσους γωνίας περιέχουσαι εὐθεῖαι, αἳτινες ἀπὸ τῆς ἀνὶσους γωνίας περιέχουσαι εὐθεῖαι, αἳτινες ἀπὸ τῆς ὂψεως \[περιέχουσαι\] φερομένας πρὸς τὸ κάτοπτρον κἀκεῖθεν πρὸς τὸ δρώμενον, μείζοντες οὖσαι τῶν τὰς ἲσας γωνίας περιεχουσῶν εὐθειῶν. καὶ ὃτι τοῦτο ἀληθές, δῆλον ἐντεῦθεν.

Ὑποκείσθω γὰρ τὸ κάτοπτρον εὐθεῖά τις ἡ ΑΒ, καὶ ἒστω τὸ μὲν ὁρῶν Γ, τὸ δ' ὁρώμενον τὸ Δ, τὸ δὲ Ε σημεῖον τοῦ κατόπτρου, ᾧ προςπίπτουσα ἡ ὂψις ἀνακλᾶται πρὸς τὸ ὁρώμενον, ἒστω, καὶ ἐπεζεύχθω ἡ ΓΕ, ΕΔ. λέγω ὃτι ἡ ὑπὸ ΑΕΓ γωνία ἲση ἐστὶ τῇ ὑπὸ ΔΕΒ. εἰ γὰρ μὴ ἒστιν ἲση, ἒστω ἓτερον σημεῑον τοῦ κατόπτρου, ἐν ᾧ προσπίπτουσα, ἡ ὂψις πρὸς ἀνίσους γωνίας περιέχουσιν ὑποκειμένης τῆς ΑΒ εὐθείας, μείζονές εἰσι τῶν ΓΕ, ΕΔ εὐθειῶν, αἳτινες τὰς ἲσας γωνἰας περιέρχουσι μετὰ τῆς ΑΒ. ἢχθω γὰρ κάθετος ἀπὸ τοῦ Δ ἐπὶ τὴν ΑΒ κατὰ τὸ Η σημεῖον  (213) | καὶ ἐκβεβλήσθω ἐπ' εὐθείας ὡς ἐπὶ τὸ Θ. φανερὸν δὴ ὃτι αἱ πρὸς τῷ Η γωνίαι ἲσαι εἰσίν. ὀρθαὶ γάρ εἰσι. καὶ ἒστω ἡ ΔΗ τῇ ΗΘ ἲση, καὶ ἐπεζεύχθω ἡ ΘΖ καὶ ἡ ΘΕ. αὓτη μὲν ἡ κατασκευή. ἐπεὶ οὖν ἲση ἒστὶν ἡ ΔΗ τῇ ΗΘ, ἀλλὰ καὶ ἡ ὑπὸ ΔΗΕ γωνία τῇ ὑπὸ ΘΗΕ γωνίᾳ ἲση ἐστί, κοινὴ δὲ πλευρὰ τῶν δύο τριγώνων ἡ ΗΕ, καὶ βάσις ἡ ΘΕ βἀσει τῇ ΕΔ ἲση ἐστί, καὶ τὸ ΗΘΕ τρίγωνον τῷ ΔΗΕ τριγώνῳ ἲσον ἐστί, καὶ <αἱ> λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις εἰσὶν ἲσαι, ὑφ' ἃς αἱ ἲσαι πλευραὶ ὑποτείνουσιν. ἲση ἂρα ἡ ΘΕ τῇ ΕΔ. πάλιν ἐπειδὴ τῇ ΗΘ ἲση ἐστίν ἡ ΗΔ καὶ γωνία ἡ ὑπὸ ΔΗΖ γωνίᾳ τῇ ὑπὸ ΘΗΖ ἲση ἐστί, κοινὴ δὲ ἡ ΗΖ τῶν δύο τριγώνων τῶν ΔΗΖ καὶ ΘΗΖ, καὶ βὰσις ἂρα ἡ ΘΖ βἀσει τῇ ΖΔ ἲση ἒστί, καὶ τὸ ΖΗΔ τρίγωνον τῷ ΘΗΖ τριγὠνῳ ἲσον ἐστίν. ἲση ἂρα ἐστὶν ἡ ΘΖ τῇ ΖΔ. καὶ ἐπεὶ ἲση ἐστὶν ἡ ΘΕ τῇ ΕΔ, κοινὴ προσκείσθω ἡ ΕΓ. δύο ἂρα αἱ ΓΕ, ΕΔ δυσὶ ταῖς ΓΕ, ΕΘ ἲσαι εἰσίν. ὃλη ἂρα ἡ ΓΘ δυσὶ ταῖς ΓΕ, ΕΔ ἲση ἐστί. καὶ ἐπεὶ παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, τριγώνου ἂρα τοῦ ΘΖΓ αἱ δύο πλευραὶ αἱ ΘΖ, ΖΓ μιᾶς τῆς ΓΘ μείζονές εἰσιν. ἀλλ'ἡ ΓΘ ἲση ἐστὶ ταῖς ΓΕ, ΕΔ. αἱ ΘΖ, ΖΓ ἂρα μείζονές εἰσι τῶν ΓΕ, ΕΔ. ἀλλ' ἡ ΘΖ τῇ ΖΔ ἐστὶν ἲση. αἱ ΖΓ, ΖΔ ἂρα τῶν ΓΕ, ΕΔ μείζονές εἰσι. καὶ εἰσιν αἱ ΓΖ, ΖΔ αἱ τὰς ἀνίσους γωνιάς περιέρχουσαι· αἱ ἂρα τὰς ἀνίσους γωνίας περιέχουσαι μείζονές εἰσι τῶν τὰς ἲσας γωνίας περιεχουσῶν.

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## The Translation
(This is a first draft of the translation. The English rendering will be tidied on the next passthrough.)

For since this is agreed upon by all, that in no way is nature employed fruitlessly nor does it do anything in vain,  if we do not grant that the reflection occurs at equal angles, nature does work in vain at unequal ones, and instead of the ray striking the object of vision by a short line, it will appear to the vision to go by a long line, this detection appears through a long line. For it will be found that the \[rays\] of the straight lines contain unequal angles, whichever ones the straight lines contain from the unequal angles, whichever ones are bearing \[them\] to the mirror \[and\] from there to the object, being larger than the straight lines containing the equal angles. And seeing that this is true, the following is clear.

Let the mirror be a certain straight line (AB), and on the one hand let the viewer be Γ, and on the other hand Δ as the object of view, while E is the point of the mirror, which upon striking, the vision is bent back to the object of view, and let ΓΕ connect ΕΔ. I say that the angle under ΑΕΓ is equal to ΔΕΒ. For if it is not equal, let it then be the other point of the mirror, upon striking it, the vision contains the stretching of the line AB toward unequal angles, they are larger than the straight lines ΓΕ and ΕΔ, whichever ones contain the equal angles beyond AB. For let the perpendicular be drawn from Δ on AB according to the point Η.

 (213) And let it fall on the straight line just as on Θ. It is no doubt apparent that these angles are equal to H. For they are straight. And let ΔΗ be equal to ΗΘ, and let ΘΖ and ΘΕ be connected. This is the reasoning. Seeing that is the case therefore ΔΗ is equal to ΗΘ, but also the angle ΔΗΕ is equal to the angle ΘΗΕ, and the line of the triangle ΗΕ is shared between the two triangles, and the base ΘΕ is equal to the base ΕΔ, and the triangle ΗΘΕ is equal to the triangle ΔΗΕ, and these remaining angles are equal to the other remaining angles, the equal lines extend to those. Therefore ΘΕ is equal to ΕΔ. Again therefore ΗΘ is equal to ΗΔ and the angle along ΔΗΖ is equal to ΘΗΖ, and ΗΖ is shared between the two triangles ΔΗΖ and ΘΗΖ, and therefore ΘΖ is equal to ΖΔ. And consequently ΘΕ is equal to ΕΔ, let (the shared line) ΕΓ be attached. Therefore the two lines ΓΕ and ΕΔ are equal to the two lines ΓΕ and ΕΘ. Therefore the whole ΓΘ is equal to the two ΓΕ and ΕΔ. And therefore the two larger sides of the remaining one of every triangle are shared on every side, consequently the two sides ΘΖ and ΖΓ of triangle ΘΖΓ are larger than the one \[side\] ΓΘ. But ΓΘ is equal to ΓΕ and ΕΔ. Therefore the two sides ΘΖ and ΖΓ are larger than ΓΕ and ΕΔ. But ΘΖ is equal to ΖΔ. Consequently, ΖΓ and ΖΔ are larger than ΓΕ and ΕΔ. Also, ΓΖ and ΖΔ contain unequal angles; therefore the contained unequal angles are larger than unequal angles containing them.

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## The Commentary

Coming soon...


### Bibliography
For more reading, see:

Drachmann, Aage Gerhardt. *The Mechnical Technology of Greek and Roman Antiquity: A Study of the Literary Sources.* Madison, WI: University of Wisconsin Press. 1963.

Heath, Thomas. ["Mensuration: Heron of Alexandria"](https://archive.org/details/historyofgreekma029268mbp/page/298/mode/2up) in A Greek History of Mathematics, Vol. 2, 298-354. Oxford Clarendon Press, 1921.


An introductory exercise in translating Heron of Alexandria

Riemann: 'On the Hypotheses which Lie at the Root of Geometry'

Uexküll translation: The External and Internal Worlds of Animals

The infamous Bergson-Einstein debate, translated

Catoptrica: The Greek Fragment, translated