11 Preface to Johannes Vogelin's Book on the Elements of Geometry (1536) CR III, 107–14
11 Preface to Johannes Vogelin's Book on the Elements of Geometry (1536)
CR III, 107–14
Greetings from Philip Melanchthon to Johannes Reiffensteyn1
I decided that no more fitting ornament can be added to the beginning of this book which provides access to geometry than the sign which Plato is said to have painted on the gates of his school, namely: ‘Let no one enter who is not trained in geometry (ageōmetrētos oudeis eisitō).’2 The interpretation of that saying has engaged the conjectures of many people. Some judge that Plato kept away from his school – as if they were defiled and impious – those inexperienced in geometry, the elements of which used to be taught then straight away from an early age to all those who were given a liberal education. Others transfer it to morals and believe that it meant, for students of philosophy, that they should preserve in all duties a certain moderateness and impartiality by analogy with geometry, just as in the Gorgias, when Plato rebukes the unjust opinion of Callicles, he says that the latter neglects geometry. Even though it is sufficiently clear from Plato's writings that he liked to apply geometrical examples to morals, nevertheless it cannot be doubted that, at the same time, by doing so he points out something about the order of subjects, and that he felt that those who were to undertake philosophy should be prepared in geometry. There are many most weighty reasons for his aphorism. For that art is not to be relegated to the mechanics who measure buildings, vessels or other trifling bodies, even though that practice also contains liberal teaching, and brings great benefits to life. But, for many other reasons, the philosopher needs geometry, for the beginnings of natural philosophy take their origin from there. And everywhere in all parts of natural philosophy many demonstrations are taken from that art: first those which show that the world is finite, that there are not several worlds and that no body is infinite. For these are the true beginnings of natural philosophy. Furthermore, since geometrical demonstrations are most distinct, no one without a knowledge of that art understands what is the power of demonstrations; without it no one will be
1 Johannes Reiffensteyn was the son of Melanchthon's friend, Gulielmus Reiffensteyn, to whom he had earlier dedicated the De Dialectica Libri Quatuor. Later Melanchthon also dedicated the Elementorum Rhetorices Libri II to both Johannes and his brother Albertus (CR II, cols. 542–44). Johannes Vogelin (d. 1549) was a professor of mathematics at Vienna and then at Heilbronn.
2 This phrase is discussed by Erasmus in Adages, III. iii.60.
a master of method. Therefore Plato, too, said that geometry needs to be studied for the reason that the knowledge of it leads to understanding the other arts more easily and correctly. But its most obvious benefit lies in measuring the size of the earth, of the heavenly bodies and of distances. And that is the greatest success of geometry, that it did not cleave to trifling and lower machines, but took flight to the heavens, and raised human minds, having cast off the earth, back to that heavenly abode, and showed us the wonderful construction and regulation of the world. Furthermore, it led exiled minds to their homeland and to acquaintance with the heavens, and even to the recognition of God. For that very teaching, in which the construction and the ruling of the world are beheld, has great power in strengthening worthy beliefs about God in the hearts of men.
Since, therefore, the origins of that most outstanding part of philosophy, on heavenly motions, are mainly in geometry, there is a sufficiently weighty reason why Plato admonished those who were about to undertake philosophy to add the study of geometry. I believe that this is what Plato meant by the inscription which I have repeated here, in order that Plato's authority might add some weight to our oration, since I wish to encourage the young to aspire to that art which has to be used as a guide towards many parts of philosophy. For every time the students take this book into their hands and read its Platonic inscription, they will think that they are being admonished by the voice of Plato, and indeed with the prayers and judgements of all the learned, so that they may aspire to that art because of the greatest benefits. Nor assuredly is there any doubt that the computation of measures in itself delights dispositions that are not warped, just as by nature we are captured by the comparison of numbers or the harmony of sounds. However, the greatness of this benefit must rouse and inflame noble and lofty minds to these studies, because this art opens the path to that most outstanding philosophy of heavenly things; how much dignity it has, and in what various ways it is useful for the life of men, is not in the least unclear, especially to those who are not utterly averse to studies of the true and old philosophy.
I know that these exhortations are of no avail when faced with those endowed with base minds, who do not discern the dignity of the outstanding disciplines, or pursue some more saleable arts for the purpose of gain. Plato certainly excludes them most strictly, since they are untrained in geometry (ageometrētoi), on two accounts. For they have monstrous minds, as well as reprehensibly throwing into disorder the geometrical proportion
since they do not concede the arts their dignity. However, upright minds, even mediocre ones, can be stimulated by the admiration of the arts, if they are encouraged, and also if there happens to be a master who teaches them skilfully. Therefore I hope that I can provoke some to study, and I urge you, my Johannes, and those like you, that you consider first that those like yourself who aspire to true praise need to strive with all the powers of intellect and mind in order to obtain for themselves a firm and perfect knowledge which is of use to the state. For that purpose the entire circle of the arts is needed, tied and joined together, so that in the individual arts many things are to be adopted from others nearby. Indeed the two disciplines,3 of numbers and of measures, have great usefulness in natural philosophy, while they also have also brought forth the entire teaching of heavenly things.
They say that when Aristippus had lost all his possessions in a ship-wreck, and he himself had nevertheless safely reached the shore of Rhodes with a few others, he caught sight of some geometrical figures on some scaffoldings when walking along the shore. Although the sea had cast them off their journey and stranded them in an unknown place, when he had seen the geometrical figures he nevertheless bade his companions be of good cheer, saying that he had seen traces of men. And he congratulated himself and the others that they had not been stranded on a barbaric shore, and asserted that humanity towards strangers and the shipwrecked would not be lacking in these men among whom the studies of these arts were cherished. If only these traces of men which Aristippus saw there on the shore were even more numerous in the schools. For these arts have been lying abandoned and neglected for many centuries already. For the previous age lured away the young from true philosophy to the most absurd sophistries; now, after they have been driven out of the schools, one would need to take pains that pure and genuine philosophy be taught, which leads to achieving firm knowledge. For this our age shows us plainly how much the state needs excellent knowledge, because everywhere many – be it from lack of judgement, or because they cannot explain anything clearly – have spread or defend absurd and miscellaneous beliefs, from which great struggles and great discord in the Church have emerged. There will be no end to these ills unless the young are called back to the true and erudite system of studying. Those who lead the Church and the state would truly need to apply themselves to that task. But the same concern is also a matter for
3 Reading scientiae for scientia.
our office, that is, for those who teach or study in the schools. For our office also pertains to the state. We are put in our position by divine providence in order to preserve and spread a knowledge useful for humankind. God demands this diligence from teachers and students in equal measure. Therefore the young should consider that they have a duty towards God to strive for firm and complete knowledge which will be of use to the Church and the state. As we bring our minds to the churches, we need to bring them to the school, that is, in order that we come to know divine things there and disclose them to others. If anyone comes to the school only in order to take away from there some particular discipline that he could direct to gain and idle show, he should know that he defiles the most holy temple of knowledge. In the same way, if the young understand their duty, if they know in what spirit they need to busy themselves with their studies, we shall easily obtain from those whose minds are not monstrous that they understand all the arts correctly and in order, and that they attempt to acquire not the empty shadow of erudition, but true knowledge.
Some are deterred by the difficulty of the subject. But these deliver the verdict before they examine, which is most unjust; before tasting the first principles, they reject and condemn the entire study. Certainly the beginnings can be understood without great trouble, and they are useful in life and in many arts. They would at all events need to know these first before they delivered a verdict on the difficulty. Furthermore the order, which is most convenient, particularly in geometry, alleviates the toil and adds much light. Finally, demonstrations are taught everywhere; although in the most advanced part of the art, as if in the pediment, they are far removed from our eyes and our sight, like cities which we see from afar, nevertheless in the other parts, being more exposed to the eyes, they contain much less difficulty. It is the utmost sloth to reject the study before you have made the attempt. And it is unjust weakness of the mind not to want to undertake any toil in studying, since it is a certain warfare to be engaged in education, and the state entrusts us with the care and preservation of the highest things which cannot be protected without the keenest striving of the minds.
Therefore both the dignity of the arts themselves and the public good should rouse us; and let us remember that to these studies of virtue we need also to add fortitude, which does not allow the mind to languish in sluggishness, and which makes its way by force, so to speak, through all difficulties. Noble and heroic natures which are urged to these highest arts about heavenly things by some divine inspiration and enthusiasm (enthousiasmōi)
grasp these arts easily, just as those who have a natural talent for poetry quickly grasp the measures of syllables and feet. Nevertheless, as in other arts, even mediocre minds will achieve much through zeal and diligence. And if some do not dedicate themselves completely to this study, they have need, nevertheless, of a knowledge of the first principles of geometry, in order to shape their judgements and understand many topics which are common in Aristotle and in other famous authors.
Aristotle depicts justice most beautifully, in the fifth book of the Ethics, by these geometrical figures, and he distinguishes its forms by comparing them to arithmetical and geometrical proportions. And in that passage the precepts are taught that are necessary for those who desire to judge learnedly about the causes of the laws. A comparison taken from arithmetic and geometry sheds most light on those. However, since the commentators whose books have come down to us did not understand that comparison they not only obscured, but utterly falsified, Aristotle's entire opinion, just as if someone had sprinkled and covered with dirt and mud an outstanding painting by Apelles. Furthermore, it is not only shameful for the commentator, but also irksome for other readers to get stuck in such a passage as if in mud4 and to be defrauded of the author's opinion. For Aristotle most sagaciously establishes two forms of justice, one of which orders the ranking of persons in choosing magistrates in positions of authority, in a city and in families. The other rules not only contracts, but any exchange of things, such as merchandise, damage, harm and penalties. Then surely, when it is explained why in exchanges the arithmetical proportion is needed, but in the other kind of justice – in choosing magistrates – geometry is applicable, the causes of justice become very clear. Aristotle took this comparison from Plato, who states with the greatest elegance and dignity that equality must be achieved in the cities, because equality creates mutual love: as one says: ‘Same is dear to same (ison isōi philon)’.5 However, he says that arithmetical equality is troublesome in positions of authority and in choosing magistrates, while geometrical equality is beneficial for cities. For it is geometrical equality when degrees are established and selection is employed, so that the highest commands are bestowed upon the best and most sagacious citizens in proportion, and the individual citizens understand which part of the public duty they support, and stay in their rank and
4 People are said to be ‘stuck in the mud’ when they are involved in troublesome business from which they cannot extricate themselves: see Erasmus, Adages, I.ii.lxxxi.
5 Cf. Erasmus, Adages, IV.vi.67.
do not upset the proportion. Plato proclaims this arrangement by saying that this geometry is divine, and that the cities then become happy indeed when God imparts to them a small part of that geometry. Then he adds that whatever is good proceeds from that geometry – just as if in the Church the highest authority belonged to the best and most learned, and individuals understood and performed their duty by their rank, and each one, as one says, adorned his own Sparta, and the inexperienced yielded to the opinions of the learned. What could be happier for the Church than if it were established by that geometrical proportion which prevents both tyranny and the wantonness of the people? For in a tyranny there are no degrees, but all the good are oppressed to an equal degree. In a democracy arithmetical equality rules, in consequence of which all the lowest men reach the highest commands without selection, and the condition is achieved which Achilles disparages most sharply in Homer, when he says that he does not want to be in a country in which there is no distinction between good and bad citizens: ‘The bad and the good would be held in equal honour’ (en d'iēi timēi ēmen kakos,ēdē kai esthlos) [Iliad IX. 3 19]. What could be more desirable for all those who are good than if the geometrical proportion – which prevents both tyranny and the wantonness of the people – governed the Church Synod? Indeed, it is not without reason that the most learned men have been delighted by geometrical analogies. For they appear before us like paintings. Therefore, since they are understood, they greatly illuminate discussions, and inform us of many things that are worthy of admiration.
Therefore that reason, too, should attract the young to getting to know the first principles: they see that great men loved these figures, and that their writings cannot be understood without having tasted these arts. For although there are much greater benefits, of which I have spoken a little while ago, for noble minds it nevertheless also adds an incentive if they love the ideas of great authors as if they were the most precious gems, and desire to grasp their power completely. These examples show that it is not absurd to apply the maxim which Plato wrote on the entrance to the school – ‘Let no one enter who is not trained in geometry’ (ageōmetrētos oudeis eisito) – to morals. He excludes from discipleship those who upset the geometrical proportion, who neither understand nor preserve the degrees of honourable duties, and who rush, without law and inconsistently, wherever their impulse carries them. For that saying of Aeschylus is most true, that each man is suited to a different kind of city (allon allēi pros polei tetagmenon) [Plato, Republic VIII, 550c]. Therefore, just as these wild and barbarous
minds are not suited for the philosophical city because they neither admire the arts nor can they be taught, so, on the other hand, those who are endowed with moderate minds are suited for the philosophical city, because they can be encouraged to cherish these best studies. Such men Plato invites, and at the same time he shows which disposition is capable of philosophy, what morals are appropriate for these studies, what minds can be roused to the love of philosophy and what kind of teaching is needed at the beginning.
Therefore, when the students read that Platonic inscription – ageōmetrētos oudeis eisitō – they should remember that they must take upon themselves geometrical equality in morals, and join the study of geometry to the other arts. Either is a great distinction and has to be striven after for many reasons.
But you, my Johannes, have been given by the favour of God a mind that is both moderate by nature and – as one can see from your drawings – talented for geometry, and clearly suited to the philosophical city. I hope that, roused by your father's and our encouragement, you are aflame with the desire to know that most beautiful teaching about the motions and effects of the heavens. The step towards this needs to be made through geometry. You will believe that you are also doing something pleasing to God by bestowing care upon these studies. For it is said most pleasantly by Plato that God always practises geometry (theon aei geōmetrein) [Plutarch, Moralia VIII.ii.718c–720c], that is, as I interpret it, He governs everything, and rules by a most firm law the heavenly courses and all of nature. Therefore without doubt He approves of the studies of those who by, observing the paths of these courses, so to speak, acknowledge and worship Him as the Ruler. Jakob Milich joined together, by a useful decision, the books on arithmetic and geometry most suited for schools, written by outstanding masters in the university which for some centuries was the main abode of these arts. For in Vienna the philosophy of celestial phenomena was almost reborn, promoted by Peuerbach,6 and now they maintain its possession, as if by inheritance, with great success, and there it is made illustrious by other men as well as by Johannes Vogelin, the author of this book on the elements of geometry, which he describes with the brevity and arrangement which he hoped would be most useful to the students. May Christ guide your studies and those of the rest of the young to the good of the state. In the month of August, in the year 1536.
6 See below, note 3 to oration 12, p. 107.