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I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

Portraits from Memory.

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# The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler

## Precession of the Equinoxes

D'Alembert's strongest case was the one against Euler's paper on the precession of the equinoxes and the nutation of the earth's axis [11]. An English translation of this article by Steven Jones is available by clicking here for the html version and here for the pdf version. The precession of the equinoxes is a phenomenon that has been known since classical times. The earth's axis is not in fact stationary and instead traces out a large circle with respect to the fixed stars, rather like a top spinning on an oblique axis. The period of the precession is about 26,000 years and it will significantly alter the location of the north celestial pole in the millennia to come. In 1748, the British Astronomer Royal James Bradley announced his discovery of another disturbance in the earth's axis of rotation, a nodding motion or "nutation" with an 18 year cycle. D'Alembert had set himself the task of explaining both phenomena in strictly mechanical terms, as a consequence of Newton's inverse-square law of gravitation. He eventually cracked the problem, and published his book-length solution [14]  in the middle of 1749.