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Those demonstrate the necessity, dignity, and utility of the mathematical disciplines. Plato and Aristotle have adopted these disciplines for a method of contemplating and doing. And certainly the best evidence of this is the Timaeus of Plato and the Physics of Aristotle. They bring forward the light of philosophy by means of mathematics itself, etc. Truly Aristotle’s whole account about motion and rest, about time and the heavens, and also about the development of animals—and indeed his entire physical discussion—abounds not so much in examples, as in geometrical foundations. Indeed by the middle of the first book of Physics Aristotle brings up Antiphon’s “squaring of the circle,” so that he may reject it. In the second book he discusses the two right angles in a triangle, which he also discusses in his Posterior Analytics. In the third book he mentions some things about building gnomons, and then the remainder concerns infinity of size, motion, and time. Most people introduced to the ideas of Aristotle lack solid understanding of these books, because they have never deeply perceived the mathematical disciplines. And in the book On the Heavens, because a diameter is not comparable to a side, Aristotle discussed a sphere constructed around eight pyramids, and so pyramids, and so triangles. His book Meteorology is full of mathematics. The same must be said regarding Metaphysics. Indeed Book 12 of Metaphysics considers whether mathematics is a real thing, whether numbers are real things, and whether mathematical ideas are the most fundamental of all. Thus a certain knowledge of mathematics is a necessity. Mathematics also pertains to Theology. For example, settling the date of the celebration of Easter—and of the rest of the “moveable feasts,” as they are called—was a concern both at the ancient synod at Nicea and at the most recent synod at Trent. The order and management of the whole Christian Republic is arranged according to that date. I shall not digress to those things which occur everywhere in the scriptures, regarding the stars and the heavens, the measurements and architecture of the temple of Solomon, and countless other things. If we consider Medicine, we find Galen stating that a doctor who is ignorant of the timing and duration of proper treatments must not treat the sick. By such ignorance a doctor may bring a patient to ruin, rather than to the health and soundness the patient might expect. Likewise a farmer who ignores the proper timing of grafting, transplanting, and sowing, will usually experience want and require charity. Plutarch reports in his Life of Marcellus how Archytas and Eudoxus added variety to geometry by removing it from the realm of pure mental exercise and bringing it into the realm of real and practical things, which are now found in Aristotle’s Mechanics. The fruits of such practical applications of mathematics are many. Archytas created a flying wooden pigeon. Archimedes and Posidonius constructed working mechanical models, or planetaria, that replicated the movements of the Sun, Moon, and planets—this is reported by Cicero, who notes that in making these planetaria they replicated the action of God in building the universe, as Plato describes in the Timaeus. In more recent times the Nürnberg artist, Albrecht Durer, illustrated his fly and his eagle with geometrical wings. Claudius Gallus constructed for the gardens of Cardinal d’Este elaborate mechanical birds, driven by hydraulic action. Small copper birds would sing and move, until a little mechanical owl appeared, and then when it departed, they would resume their activities. They were so realistic that a person who declared them fakes would seem more temerarious than a person who claimed them real birds would seem credulous. Other practical fruits of mathematics relate to measurement. A single measuring rod, used to measure distances, can also be used to determine the areas of surfaces and the volumes of bodies. Using a simple measuring rod, any geometer can describe buildings, lands, seas, the movements of the heavens, the risings of stars, and so forth. But these things are not all that is encompassed by the discipline of Mathematics. Plato wrote:
In dealing with encampments and the occupation of strong places and the bringing of troops into column and line and all the other formations of an army in actual battle and on the march, an officer who had studied geometry would be a very different person from what he would be if he had not.
Indeed, there are many military applications of geometry. In the Roman army, a centurion’s flag served as a point around which a circular or rectangular formation of troops would be established, depending upon the circumstances of battle. Geometry guides the construction of bridges and ships, the channeling of water, and the movement of cavalry among foot soldiers. It can be used in both attack and defense—in the construction of both siege engines and defensive ramparts. Geometry has influenced the outcome of a remarkable variety of battles. A small group of Caesar’s soldiers broke through the ranks of a vast army by means of a wedge formation, and escaped unharmed. Likewise three hundred legionaries held back another vast army for hours through the use of a circular formation. At Syracuse, Archimedes constructed such defenses against Marcellus that Marcellus called him “this Briarian engineer and geometrician [who] hath with shame overthrown our navy, and exceeded all the fabulous hundred hands of the giants, discharging at one instant so many shot among us.” Zonaras reports that Proclus by means of mirrors fashioned to collect and concentrate light from the sun, “burnt the fleet of Vitellius, at the siege of Constantinople, in imitation of Archimedes, who set fire to the Roman fleet at the siege of Syracuse.” And Archimedes and Proclus are but two of the people who have used geometry for military purposes.
(excerpted from Poem and Letter to the Reader)
