10 Preface to arithmetic (1536) CR XI, 284–92
10 Preface to arithmetic (1536)
CR XI, 284–92
Preface to arithmetic read by Georg Joachim Rheticus1
I believe that in the beginning I need to expound to you the reason why I begin to teach publicly, so that no one thinks that I have come forth in this hall inconsiderately by my own decision. For I am rather shy by nature, and I cherish most those arts that love hiding-places and do not earn applause among the crowds; furthermore, I see not only what abundance of knowledge [i.e. of things to be known] there is, but also how much dexterity and fertility of intellect is needed, especially for treating of these subtle arts, and I am indeed conscious of my stupidity. Therefore I did not strive to be here, and for a long time I objected to our teachers who encouraged me to teach in public the precepts of the arts with which I have busied myself in an unremarkable way, in particular at this time, when the other lectures in mathematics have ceased by the death of that most learned man, Johannes Volmar.2 However, they entreated me to take into account, in this matter, not only my friendship for them but also the benefit for the young, as it is greatly profitable that these elements be assiduously inculcated and practised in the schools. Thus the teachers' authority prevailed, as well as their excellent disposition towards education. I saw that they did not confer this duty upon me out of private favour but in order to be mindful of the young, and to attract them, as far as possible, to the arts that are the true beginning of philosophy.
Therefore, while I did not want my effort to be lacking for the use of the state and of the young, they nevertheless urged me to promise that I would take the risk. I entreat you, therefore, to consider that because of this duty I have come forth in public not out of arrogance, but out of great trust. Moreover, I shall repent my willingness less if I recognise that your attendance and diligence respond to the excellent disposition of the teachers, who imposed this rôle on me for the sake of your benefit. It will therefore
1 Georg Joachim Rheticus (1514–74) was promoted to MA at the University of Wittenberg on 27 April 1536. He is best known for having persuaded Nicholas Copernicus to publish the De revolutionibus: see K. H. Burmeister, Georg Joachim Rhetikus, 1514–1574, Eine Bio-Bibliographie, 3 vols. (Wiesbaden, 1967–8).
2 Johannes Volmar (d. 1536) was a lecturer in mathematics at the University of Wittenberg from 1518 to 1536. His notes on astrology and astronomy survive at the University Library of Jena: see J. C. Mylius, Memorabilia bibliothecae academicae Jenesis (Jena, 1746), pp. 348, 405f.
be a sign of your sagacity and humanity to take my effort in good part. For I hope to satisfy you as far as teaching the precepts themselves is concerned, but I ask your forgiveness if you find something lacking in my speech or my delivery. I admit that I am hampered by embarrassment and fear, and that by nature I am not suited to theatrical performance. However, it is a characteristic of an effeminate and peevish mind to shrink from these arts because of these inconveniences, and thrust away the study of them, even if a teacher is expert in his own discipline. Indeed, even if I do not assume anything for myself, I shall nevertheless teach these precepts with faith and diligence in the hope that they will be grasped and understood.
I have expounded the reason why I have come forth in public, which, I hope, will free me from the suspicion of arrogance with the good and the learned. Now I shall speak of my opinion, or rather of that of our teachers. They enjoined me to teach the elements of arithmetic and geometry and, since the nature of these arts demands it, I shall begin from arithmetic. Would that I were so effective, both in authority and in eloquence, as to rouse and inflame the young to the study of these arts. For what could be more desirable for the teacher than to have listeners who are kindled and aflame with the desire to learn? But since our teachers daily encourage the students to learn these arts, and their benefit is evident, I hope that most of those who are endowed with superior minds are sufficiently inspired to cherish these studies.
The Greeks tell the story of Pallas [Athena] who, when Mercury had given her the flute he had recently invented, walked to a stream and began to play it. Although she was charmed by the sound, she nevertheless threw away the flute immediately when she saw in her reflection in the water that her face was ridiculously deformed by the blown-up cheeks. The Satyrs who had approached, attracted by the novelty of the sound, immediately took up what she had thrown away, and thus the flute, spurned by Pallas, began to be the favourite of the Satyrs. In the same way let ignoble minds rejoice in other rustic arts, or rather divulge some fragments of the arts, either for the sake of gain or in order to win over public opinion. But it is characteristic of the honourable mind not to love anything more ardently than truth, and, inspired by this desire, to seek a genuine science of universal nature, of religions, of the movements and effects of the heavens, of the causes of change, not only of animated bodies but also of cities and realms, of the origins of noble duties and of other such things. If we are held by admiration and love for this perfect science, then we also need to value
those elements of numbers and measures that provide access to the other parts of philosophy, although even on their own they are of great nobility and benefit.
For you see how great is the use of arithmetic in economy, and Aristotle writes in the Republic3 that there are some Thracians who cannot count beyond four. I ask you whether you think that the management – I am not even saying of a large market or of mines of metals, but of some medium-sized household – can be entrusted to such men? Would we think that such men can unravel and settle an even slightly intricate computation? By no means. But when it comes to long and obscure computations, all those who are deprived of the assistance of this art are similar to those Thracians. This art is necessary not only on the market and for metals for coining money, but in many other public and private computations. Those who busy themselves for the state must know the calculation of finances. In the courts of justice there often occur problems which require learned calculations. Therefore it is not sufficient to know the numbers through the kindness of nature, but the art has to be added without which certainly no difficult calculations can be settled. Therefore in Plato [Laws 819ff.] Socrates makes it one of the laws that the citizens be compelled to learn arithmetic. But what is the use of speaking at length about a known fact? For there is no one who does not see that in every life the greatest use is made of arithmetic – unless he is more stupid than those Thracians who cannot count beyond four. And these everyday benefits should attract equally all the young to the study and the practice of this art, for many eventualities can arise in which arithmetic can be of great use to the individual. Therefore the Romans, too, took great care so that their children grasp arithmetic as well, together with the first steps in reading and writing. You will remember that verse of Horace [Art of Poetry, 325–8]: ‘Roman boys learn, in long computations, to divide the as4 into a hundred parts. ‘‘Let the son of Albinius answer: If one ounce is taken from five-twelfths, what remains?”’ The benefit of geometry, too, is manifold, not only in building and in measuring vessels, but also to a statesman for cosmography, for understanding the distance between places. But I forsake these everyday benefits.
Those who busy themselves with studies and strive for complete knowledge should see clearly that there is no access to the science of celestial things except through arithmetic and geometry. Arithmetic is so important
3 In fact, Aristotle's Problems, 911a2.
4 An as is a copper coin of small value, a penny.
in the science of celestial things that almost everything in that science is accessible to even a mediocre arithmetician; he can certainly attain to a large part of that science without any trouble. See by what little toil you can obtain a great reward for your work! Nothing is easier than to teach these notions, as they call them, of the art of counting. If one knows them even imperfectly, one can immediately grasp almost all of astronomy without any difficulty.
In the Phaedrus [246c] Plato invents two kinds of soul, one of which he says is winged, the other has lost its wings. Then he says that those that have wings fly up to heaven and delight in meeting and conversing with God, and in the most beautiful sight of the heavenly ways, and he says that they contemplate the causes of all changes in inferior nature, in the air, in animated bodies, in the studies and manners of men and in the various vicissitudes of realms and cities. And, flying throughout all of heaven, these souls are charmed by the beauty of divine things and by the sweetness of the knowledge and of the virtue of that admirable order, and they wish to enjoy this one pleasure forever. They do not burden their souls with obscene pleasure, which perturbs the harmony of virtue in the souls and casts darkness in their way, so that they cannot behold the heavenly things.
The souls whose wings have fallen off wander about on the ground and desire impure pleasures from earthly things, and they do not behold that most beautiful light of heavenly things. Even if Plato thinks of the wings as heroic impulses of the minds, it is nevertheless not these impulses alone that lift up the minds, but the arts are also needed by which these impulses are raised up. Consequently, the wings of the human mind are arithmetic and geometry. If someone endowed with an intellect that is not mean attached these to himself, he would easily enter heaven and would wander freely in the heavenly company, and enjoy that light and wisdom.
Therefore those who are endowed with a mind that is not unnatural, and who desire to behold the things that are best and most admirable and worthy of knowledge, should attach those wings, that is, arithmetic and geometry, to themselves. Carried up to heaven by their help, you will be able to traverse with your eyes the entire nature of things, discern the intervals and boundaries of the greatest bodies, see the fateful meetings of the stars, and then understand the causes of the greatest things that happen in the life of man. I believe that noble minds can be attracted by such great benefit, and roused to learn and love those arts. For I know that you are certainly convinced that the science of celestial things has great dignity and
usefulness, and that, as is proper, your ears and your souls shrink from the absurdities of Epicurus, who derides astronomy and thinks idly that the sun is vapour, which in the morning is kindled by its movement, and then either burns up or is extinguished in water. In the same way he also says that the stars are little clouds set alight by their fall. It is unworthy of a man to declare these monstrosities as the truth, much more so for a philosopher. There has not been a people so barbarous as not to feel that the stars are true and enduring works of God, and that they are carried by firm laws, because of some great purpose. Therefore it is an insult against God to invent this confusion of things, that the sun is a vapour set alight by its fall and carried by its fall, as Lucretius says: ‘whither their food calls each and invites them as they go, grazing their fiery bodies all over the sky’.5 The authority of the most learned men who refuted these absurdities with very serious arguments must have greater value for us, as well as the divine oracles in the Holy Scriptures, from which it is impiety to dissent. These prove clearly that the sun, the moon and the other stars are true and enduring works of God, and the Holy Scriptures add for what purpose the heavenly lights are made: Let them be signs and set apart the seasons and the years [Genesis 1:14]. Although this description is rather short, it nevertheless contains great things and approves the study of astronomy. For, leaving out the rest, if the sun is made in such a way that it brings about and rules the year, the observation of the course of the sun is necessary; for without the observation of its movement there is no distinction between the seasons and the years. It is therefore evident that it is considered good by God and that the observation of the movements of the heaven is enjoined. For, apart from bringing great benefit to universal life, this most beautiful order of movements also reminds us that nature does not exist by chance, but is created and governed by an eternal mind. Hence it strongly confirms in the mind worthy beliefs about God and providence. Consequently, among the philosophers only the Epicureans were ungodly (atheoi), because they did not want to behold these illustrious proofs of God, that is, the most firm laws of movements and this wonderful harmony. It is therefore proper for good minds to love this most worthy science, and, as far as possible, to put some effort into the examination of these great things. And those who are even moderately trained in arithmetic will easily understand many things.
5 Lucretius, On the Nature of Things, v. 524f., though Melanchthon s text is slightly different from the modern text.
I shall therefore return to arithmetic, for that is what I specifically intended to talk about. After having recalled to mind all its benefits, which are, however, hardly indistinct, I have reckoned that something needs to be added briefly on its easiness. I know that the young are deterred from these arts by their reputation for difficulty. But as far as the beginnings of arithmetic are concerned, which are taught in schools and are employed for everyday use, they err greatly if they think that they are very difficult. The art originates from the very nature of the human mind and has very sure demonstrations. Therefore the beginnings can be neither unintelligible nor difficult; on the contrary, the first precepts are so clear that even boys can acquire them, because the whole thing originates from nature. Then the precepts of multiplications and divisions require a little more diligence, but the causes can be grasped quickly by the industrious. This art, like all the others, requires practice and habit. For it is very true what Anaxagoras says, that the hand is the cause of wisdom [Testimonies, fr. 102.1], meaning that without practice and experience the arts cannot be correctly understood. Therefore, wherever there is some difficulty, remember that there is no art that does not have some difficulty. And you are indeed mistaken if you believe that the practice of composition is easier than these beginnings of arithmetic; the schools are not made and established for the purpose of granting leisure to the idle for enjoying foolish pleasures, but they require studies, attention and sharp contest of minds. Nor do I deny that many things are sought out and observed by arithmeticians which elude the grasp of the inexperienced, for all the arts grow by practice. But I am talking about the beginnings that are usually taught to you and are employed for use. Therefore let the young thrust away the idea of difficulty, and approach the learning of this art with an open mind and some hope; if they do so, the sweetness of the things will engage them, and they will feel the difficulty diminish by the very connection of the things. I have expounded what benefits are to be sought from arithmetic, and I have also spoken about its efficacy. What remains are other benefits, but more obscure ones, of which I shall not say much. It is most true that, once one knows arithmetic, the other arts can be understood more easily. For arithmetic is among the first arts, which teaches us to distinguish and divide between one and many; it shows the order of things, and teaches us that confused things need to be unravelled and distinguished. These are the beginnings of reasoning in men. Therefore the understanding of numbers is foremost, and I believe that Pythagoras noticed that when he defined the mind as a number, for
this means that the mind is capable of reasoning, it discerns things and perceives order. These are the main characteristics of the mind and its foremost functions. Therefore he rightly defined the mind as a number, that is, as something that counts. This is the reason why the ancient Greeks, too, taught arithmetic second after grammar, then in the third place they ranked dialectics and in the fourth geometry. For they saw that dialectics takes its beginning from arithmetic, and the practice of multiplication and division prepares the intellect for syllogisms in an excellent way. In the same way the power of demonstration can be better understood once one knows arithmetic, because it has extraordinarily distinct demonstrations.
Besides, there are many passages in the [ancient] authors which cannot be understood without this art. The Pythagoreans used arithmetical terms figuratively and transferred them to physics and to moral philosophy. For this is how Pythagoras defined God: God is that by which the greatest number exceeds the next number. Thus he meant to say that God is one, and indeed the highest nature, for he understands the unit of 10 as the highest number, and the one next to it is the unit of 9. The projection beyond 9 is unity; thus they wrapped all of philosophy in numbers.6
Even if it is better to teach the arts in the proper sense and not to wrap them all in figures and riddles, nevertheless other philosophers, too, who clearly intended to teach philosophy, employ examples from arithmetic which it is unseemly not to understand. When Plato is saying that the state changes because of heavenly causes which effect certain vicissitudes of affairs in cities and realms, he says: just as nature resides in tones, when the four thirds are joined to five, that is when a quarter (dia tessarōn) is added to a fifth (dia pente), that means when the harmony of all (dia pasōn) is complete. In just that way, when the period of realms has passed away, states fall and change. He adds the reason that, just as when plane squares are added up to form solids until nature can go no further, many dissimilar numbers are brought forth, so ignoble citizens who, descended from more outstanding ones, have nevertheless become masters of the state, little by little overturn the state until the overthrow. Not even these designations can be understood without arithmetic.
But these benefits are less weighty, and the ones that I mentioned above are great, namely that this art is needed in many other affairs in life and,
6 According to Aristotle (Metaphysics I.v.3), the Pythagoreans considered the number 10 to be perfect, comprising the whole nature of numbers and tried to explain the universe with their number mysticism. For further details, see G. E. R. Lloyd, Early Greek Science: Thales to Aristotle (London, 1970), pp. 24–35.
without it, there is no access to that most excellent part of philosophy dealing with heavenly matters. That is the great and weighty reason which must kindle superior natures to this art, because those who seek complete knowledge worthy of great men must strive with all the powers of the intellect7 to add to the other general disciplines the part dealing with the movements and effects of the heavens. We shall rightly call happy the man who has comprehended all of philosophy in his mind, and understands the hidden causes of most changes in nature and in the lives of men, the sources of virtue and of honourable duties – as Virgil, too, said most wisely: ‘Happy is he who could know the causes of things’ [Virgil, Georgics II.490]. I have spoken.
7 Reading ingenii viribus for ingeniis viribus.