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the Arabs were enchanted by Ptolemy, and envisioned no grander cosmos. Aristarchus’ treatise on astronomical distances was translated in the early tenth century by a Syrian-Greek scholar named Questa ibn Luqa, and an Arabic secret society known as the Brethren of Purity published an Aristarchian table of wildly inaccurate but robustly expansive planetary distances, but otherwise little attention was paid to the concept of a vast universe. The generally accepted authority on the scale of what we today call the solar system was al-Farghani, a ninth-century astronomer who, by assuming that the Ptolemaic epicycles fit as tightly as ball bearings between the planetary spheres—“there is no void between the heavens,” he asserted—estimated that Saturn, the outermost known planet, was eighty million miles away. 12 Its true distance is more than ten times that.
    The Islamic devotees of Ptolemy, however, inadvertently undermined the very cosmology they cherished, by transmuting Ptolemaic abstractions into real, concrete celestial spheres and epicycles. So complex and unnatural a system, palatable if regarded as purely symbolic, became hard to swallow when represented as a genuine mechanism that was actually out there moving the planets around. The thirteenth-century monarch King Alfonso (“the Learned”) of Castile is said to have remarked, upon being briefed on the Ptolemaic model, that if this was really how God had built the universe, he might have given Him some better advice.
    But that was many long, dark centuries later. The last classical scholar in the West was Ancius Boethius, who enjoyed power and prestige in the court of the Gothic emperor Theodoric at Ravenna until he backed the losing side in a power struggle and was jailed. In prison he wrote
The Consolation of Philosophy
, a portrait of the life of the mind illuminated by the fading rays of a setting sun. There,Boethius contrasts the constancy of the stars with the unpredictability of human fortune:
    Creator of the starry heavens,
Lord on thy everlasting throne,
Thy power turns the moving sky
And makes the stars obey fixed laws
…………….
All things thou holdest in strict bounds,—
To human acts alone denied
Thy fit control as Lord of all.
Why else does slippery Fortune change
So much, and punishment more fit
For crime oppress the innocent? 13
     
    In words the Greek Stoics would have appreciated, the muse of philosophy upbraids Boethius for his self-pity. “You are wrong if you think Fortune has changed towards you,” she tells him. “Change is her normal behavior, her true nature. In the very act of changing she has preserved her own particular kind of constancy towards you.” 14
    In Boethius, the universe of Ptolemy is reduced to a symbol of resignation to the vicissitudes of fate:
    Consider how thin such fame is and how unimportant. It is well known, and you have seen it demonstrated by astronomers, that beside the extent of the heavens, the circumference of the earth has the size of a point; that is to say, compared with the magnitude of the celestial sphere, it may be thought of as having no extent at all. The surface of the world, then, is small enough, and of it, as you have learnt from the geographer Ptolemy, approximately one quarter is inhabited by living beings known to us. If from this quarter you subtract in your mind all that is covered by sea and marshes and the vast area of desert by lack of moisture, then scarcely the smallest of regions is left for men to live in. This is the tiny point within a point, shut in and hedged about, in which you think of spreading your fame and extending your renown. 15
     
    Boethius was executed in 524, and with the extinguishing of that last guttering lamp the darkness closed in. The climate during the Dark Ages grew literally colder, as if the sun itself had lostinterest in the mundane. The few Western scholars who retained any interest in mathematics wrote haltingly to one another, trying to recall