Appendix: Selection of original texts A.1 Plato: “The Republic” Following his well known “Allegory of the Cave” Plato developed the educa- tional program for the prospective people responsible for the state in the 7th book of “The Republic” – by letting Socrates explain his ideas in a dialogue with some partners. It is meant to enable them to recognise the unchange- able being and the truth based on this behind all perishable appearances of becoming and decaying. He argues that studying the four mathematical sciences is indispensible for this. Here, some extracts from the conditions Socrates lists: “And our guardian is both warrior and philosopher? Certainly. Then this is a kind of knowledge which legislation may fitly prescribe; and wemustendeavourtopersuadethosewhoareprescribetobetheprincipal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being. That is excellent, he said. (...) Let this then be made one of our subjects of education. And next, shall we enquire whether the kindred science also concerns us? You mean geometry? Exactly so. Clearly, he said, we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the difference whether a general is or is not a geometrician. Yes, I said, but for that purpose a very little of either geometry or calculation will be enough; the question relates rather to the greater and more advanced part of geometry — whether that tends in any degree to make more easy the vision of the idea of good; and thither, as I was saying, all things tend which compel the soul to turn her gaze towards that place, where is the full perfection of being, which she ought, by all means, to behold. True, he said. Then if geometry compels us to view being, it concerns us; if becoming only, it does not concern us? (...) And suppose we make astronomy the third — what do you say? © Springer Basel 2015 565 C.J. Scriba, P. Schreiber, 5000 Years of Geometry, DOI 10.1007/978-3-0348-0898-9 566 Appendix I am strongly inclined to it, he said; the observation of the seasons and of months and years is as essential to the general as it is to the farmer or sailor. I am amused, I said, at your fear of the world, which makes you guard against the appearance of insisting upon useless studies; and I quite admit the diffi- culty of believing that in every man there is an eye of the soul which, when by other pursuits lost and dimmed, is by these purified and re-illumined; and is more precious far than ten thousand bodily eyes, for by it alone is truth seen. (...) Then, I said, in astronomy, as in geometry, we should employ problems, and let the heavens alone if we would approach the subject in the right way and so make the natural gift of reason to be of any real use. (...) The second, I said, would seem relatively to the ears to be what the first is to the eyes; for I conceive that as the eyes are designed to look up at the stars, so are the ears to hear harmonious motions; and these are sister sciences — as the Pythagoreans say, and we, Glaucon, agree with them? Yes, he replied. (...) Now, when all these studies reach the point of inter-communion and connec- tion with one another, and come to be considered in their mutual affinities, then, I think, but not till then, will the pursuit of them have a value for our objects; otherwise there is no profit in them.” [Plato: The Republic (On shadows and realities in education), Plain Label Books: translated by Benjamin Jowett, 1930, p. 431 – 445] A.2 Archimedes: Introduction to treatise “On Spirals” Differently to the other scholars, who worked at the Musaeum in Alexan- dria and could converse with their colleagues there about their research, Archimedes worked very isolated in his hometown Syracuse on the island of Sicily. In contrast to, e.g., Euclid, who did not start his “Elements” with any kind of explanatory introduction at all, Archimedes stated details on the motives, which had inspired him to his investigations, in accompanying let- ters to some of his texts, which he sent to the mathematicians in Alexandria. After his friend Conon had died, he addressed several letters to Dositheus, since he had heard that he had also been Conon’s friend and, furthermore, was a experienced mathematician. “Archimedes to Dositheus greeting. Of most of the theorems which I sent to Conon, and of which you ask me from time to time to send you the proofs, the demonstrations are already before you in the books brought to you by Heracleides; and some more are also contained in that which I now send you. Do not be surprised at my taking a considerable time before publishing these proofs. This has been owing to my A.2 Archimedes: Introduction to treatise “On Spirals” 567 desire to communicate them first to persons engaged in mathematical studies and anxious to investigate them. In fact, how many theorems in geometry which have seemed at first impracticable are in time successfully worked out! Now Conon died before he had sufficient time to investigate the theorems referred to; otherwise he would have discovered and made manifest all these things, and would have enriched geometry by many other discoveries besides. For I know well that it was no common ability that he brought to bear on mathematics, and that his industry was extraordinary. But, though many years have elapsed since Conon’s death, I do not find that any one of the problems has been stirred by a single person. I wish now to put them in review one by one, particularly as it happens that there are two included among them which are impossible of realisation [and which may serve as a warning] how those who claim to discover everything but generate no proofs of the same may be confuted as having actually pretended to discover the impossible. What are the problems I mean, and what are those of which you have already received the proofs, and those of which the proofs are contained in this book respectively, I think it proper to specify. The first of the problems was, given a sphere, to find a plane area equal to the surface of the sphere; and this was first made manifest on the publication of the book concerning the sphere, for, when it is once proved that the surface of any sphere is four times the greatest circle in the sphere, it is clear, that it is possible to find a plane area equal to the surface of the sphere. [Six more problems concerning the sphere follow.] Of all the propositions just enumerated Heracleides brought you the proofs. The proposition stated next after these was wrong (...). If a sphere be cut into unequal parts by a plane at right angles to any diameter in the sphere, the greater segment of the surface will have to the less the same ratio as the greater segment of the sphere has to the less a ratio less than the duplicate ratio of that which the greater surface has to the less, but greater than the sesquialterate of that ratio. After these came the following proposition about the spiral, which are as it were another sort of problem having nothing in common with the foregoing; and I have written out the proofs of them for you in this book. They are as follows. If a straight line of which one extremity remains fixed be made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line revolves, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane. I say then that the area bounded by the spiral and the straight line which had returned to the position from which it started is a third part of the circle described with the fixed point as centre and with radius the length traversed by the point along the straight line during the one revolution... .” [Archimedes: The works of Archimedes. Translated and edited by T. L. Heath, (1897), published by Cambridge University Press, p.151 – 154] 568 Appendix A.3 Pope Gregory the Great mentions the art of land surveying One of the last mentions of the Roman art of land surveying is found in a letter by Pope Gregory the Great from July 597: Gregory to John, the Bishop of Syracuse: “To prevent disputes about secular matters estranging the hearts of the faith- ful, great care must be taken that a dispute may be settled as easily as pos- sible. We have learnt from Caesarius, Abbot of the Monastery of St. Peter at Baiae [at the Golf of Naples], that a serious dispute has arisen between him and John, Abbot of the Monastery of St.