Wednesday, October 27, 2010

St. Albertus Magnus's Commentary on Euclid's Elements

St. Albertus Magnus (1206-1280) wrote extensively on physics methodology, among other fields like theology and the experimental sciences, and is credited for discovering arsenic. He said: "The aim of natural science is not simply to accept the statements [narrata] of others, but to investigate the causes that are at work in nature" (De Miner., lib. II, tr. ii, i) and "Experimentum solum certificat in talibus" ("Experiment is the only safe guide in such investigations;" De Veg., VI, tr. ii, i). Against what today one would call "intelligent design," St. Albert says: "In studying nature we have not to inquire how God the Creator may, as He freely wills, use His creatures to work miracles and thereby show forth His power: we have rather to inquire what Nature with its immanent causes can naturally bring to pass" (De Coelo et Mundo, I, tr. iv, x).
     The Dominican Scholastic philosopher Albertus Magnus (1193?-1280) acquired the title Doctor Universalis from the breadth of his learning, which astonished his contemporaries. His fame earned him a place in the Divine Comedy, where his most famous student, Thomas Aquinas, introduces him to Dante:
Questi che m'è a destra più vicino
Frate e maëstro fummi, ed esso Alberto
È di Cologna, e io Thomas d'Aquino. (Paradiso, X, 97-99)

He who is nearest to me on the right
Was my colleague and teacher, namely, Albert
Of Cologne, and I am Thomas Aquinas.
The program of the scholastic philosophers was to use the deductive method of mathematics to demonstrate by reason the existence of Deity and to describe His attributes, to prove the immortality of the soul, to assert free will, and in general to establish thereby the truth of the Catholic religion. Their first axiom was, that this was possible. Even Russell, who considered theology nothing more than organized ignorance, could nevertheless respect the Medieval mastery of logic:
     The medieval outlook of educated men had a logical unity which has now been lost. We may take Thomas Aquinas as the authoritative exponent of the creed which science was compelled to attack. He maintained—and his view is still that of the Roman Catholic Church—that some of the fundamental truths of the Christian religion could be proved by the unaided reason, without the help of revelation. Among these was the existence of an omnipotent and benevolent Creator. From His omnipotence and benevolence followed that He would not leave His creatures without knowledge of His decrees, to the extent that might be necessary to obey His will. There must therefore be a Divine revelation, which, obviously, is contained in the Bible and the decisions of the Church. This point being established, the rest of what we need to know can be inferred from the Scriptures and the pronouncements of oecumenical Councils. The whole argument proceeds deductively from premisses formerly accepted by almost the whole population of Christian countries, and if the argument is, to the modern reader, at times faulty, its fallacies were not apparent to the majority of learned contemporaries.
     Now logical unity is at once a strength and a weakness. It is a strength because it insures that whoever accepts one stage of the argument must accept all later stages; it is a weakness because whoever rejects any of the later stages must also reject some, at least, of the earlier stages. The Church, in its conflict with science, exhibited both the strength and the weakness resulting from the logical coherence of its dogmas [Religion and Science, 12-13].
Within this program, it was the special enterprise of Albertus Magnus to interpret the works of Aristotle for his Medieval contemporaries, insofar as the writings of that philosopher could be known through the veil of Latin translations of Arabic translations of the Greek texts, which at that time covered the hearts of the learned. In so doing, he made a major contribution to culture, for Aristotle was interested in the natural world, and investigation into his activities tended to promote the rebirth of knowledge, which was accomplished in the period called the Renaissance. The labors of pople like Albertus Magnus were appreciated by Macaulay, who noticed them in one of the most famous chapters in literature:
     Whatever reproach may, at a later period, have been justly thrown on the indolence and lukury of religios orders, it was surely good that, in an age of ignorance and violence, there should be quiet cloisters and gardens, in which the arts of peace could be safely cultivated, in which gentle and contemplative natures could find an asylum, in which one brother could employ himself in transcribing the Aeneid of Virgil, and another in meditating the Analytics of Aristotle, in which he who had a genius for art might illuminate a martyrology or carve a crucifix, and in which he who had a turn for natural philosophy might make experiments on the properties of plants and minerals [The History of English from the Accession of James II, I, 9]
     Albertus Magnus is one of those personalities who are appreciated even by those whith a critical attitude towarsd the Catholic religion. For example, White, cofounder and first president of Cornell University, a declared enemy of dogmatic theology, had a sympathetic opinion of him:
First among these was Albert of Bollstädt, better known as Albert the Great, the most renowned scholar of his time. Fettered though he was by the methods sanctioned in the Church, dark as was all about him, he had conceived better methods and aims; his eye pierced the mists of scholasticism; he saw the light, and sought to draw the world toward it. He stands among the great pioneers of physical and natural science; he aided in giving foundations to botany and chemistry, he rose above his time, and struck a heavy blow on those who opposed the possibility of human life on the opposite sides of the earth; he noted the influence of mountains, seas, and forests upon races and products, so that Humboldt justly finds in his works the germs of physical geography as a comprehensive science [A History of the Warfare of Science with Theology in Christendom, I, 377].
The most recent biography of Albert, written on the occasion of the seven hundredth anniversary of his death, is by Weisheipl ["The Life and Works of St. Albert the Great". Albertus Magnus and the Sciences — Commemorative Essays].
     Albertus Magnus conceived and carried out the plan of commenting on all human knowledge by beginning with the natural sciences, proceeding to mathematics, and finishing in philosophy and theology. Thus, in his writings in the natural sciences, he refers to mathematics as the object of future activity:
     Primo complebimus, Deo adiuvante, scientiam naturalem, et deinde loquemur de mathematicis omnibus, et intentionem finiemus in scientia divina (Physica, Book I, Treatise 1, Chapter i).
     With God's help, we shall first complete natural science, and then we shall talk about all of mathematics, and we shall finish our program in divine science.

     Longum esset demonstrare, sed in geometria hoc docebitur et in astronomia, Domino concedente (Op. cit., I, 2, i).
     It would be long to prove, but, God willing, this will be taught in geometry and astronomy.

     Haec autem omnia supponenda sunt, probanda autem in libris de visu in Perspectivis, quae scientia compleri non potest, nisi primum consideremus ea quae pertinent ad geometriam (De Sensu et Sensato, Treatise 1, Chapter 14).
     All these things, however, must be supposed for now; they are to be proven, though, in the books on sight in Perspective, which science cannot be completed unless we shall first consider those things that pertain to geometry.
In his compositions on philosophy and theology, however, Albert refers to his work on geometry as at hing of the past, for he accepted the idea of Plato, that one could not competently study philosophy without having first mastered mathematics.
     Naturalibus et doctrinalibus iam, quantum licuit, scientiis elucidatis, iam ad veram philosophiae sapientiam accedamus (Metaphysica, Book I, Treatise I, Chapter i).
     Now that the natural and mathematical sciences have been elucidated as much as was possible, let us proceed to the true wisdom of philosophy.

     Hoc autem iam a nobis in geometricis est demonstratum (Ibid., I, 2, x).
     For this [sc. that the diameter and side of a square are incommensurable] has already been proven by us in the geometrical [works].

     Sicut in XV and XVI tertii geometriae nostrae demonstratum est (Ibid., III, 2, iii).
     Just as has been proven in the fifteenth and sixteenth [propositions] of the third [book] of our Geometry [sc. namely, that a tangent line to a circle intersects it at only one point].

     Sicut nos in I nostrae geometriae ostendimus (Ibid., V, 3, i).
     As we showed in the first [book] of our Geometry [sc., that two straight lines do not enclose a surface].

Albertus Magnus on Euclid's Elements of Geometry, pgs. xi-xv

Here is St. Albertus Magnus's commentary on Euclid's famous proof of the Pythagorean Theorem, Proposition 46 here:

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