Kelvin thought that he had a fairly good handle on two of these quantities. Measurements by a number of geologists have shown that while results varied from location to location, in the mean, the temperature toward the Earth’s center increased roughly by one degree Fahrenheit for every fifty feet of descent (this quantity is known as the temperature gradient). Concerning the thermal conductivity, Kelvin relied on his own measurements for two types of rocks and for sand to give him what he regarded as an acceptable average. The third physical quantity—the Earth’s deep internal temperature—was extremely problematic, since it couldn’t be measured directly. But Kelvin was not a man easily deterred by such difficulties. Putting his analytic mind to work, he was eventually able to deduce an estimate for the unknown internal temperature. The entangled intellectual maneuvering that he had to perform to achieve this result presented Kelvin at his best—and his worst. On one hand, his virtuosic command of physics and his ability to examine potential alternatives with a razor-sharp logic were second to none. On the other, as we shall see in the next chapter, due to his overconfidence, he could sometimes be completely blindsided by unforeseen possibilities.

Kelvin started his assault on the problem of the Earth’s internal temperature by analyzing a variety of possible models for the cooling Earth. The general assumption was that the Earth’s initial state was molten, as a result of the heat generated by some collision—either with a number of smaller bodies, such as meteors, or with one body of nearly equal mass. The subsequent evolution of this molten sphere depended on a property of rocks that was not known with certainty: whether upon solidifying, molten rock expanded (as in the case of freezing water) or contracted (as metals do). In the former case, one could expect the solid crust to float over a liquid interior, just like ice on the surface of lakes in winter. In the latter, the denser solid rocks forming near the Earth’s cooler surface would have sunk down, eventually forming perhaps a solid scaffolding that could support the surface crust. While the empirical evidence was scarce, experiments with melted granite, slate, and trachyte all seemed to point in the direction of molten rock contracting both upon cooling and solidifying. Kelvin used this information to chart a new scenario. He proposed that before complete solidification took place, the cooler surface liquid had sunk toward the center, thus maintaining convection currents similar to those generated in the oil in a frying pan. In this model the convection was assumed to sustain a nearly uniform temperature throughout. Consequently, Kelvin assumed that at the point of solidification, the temperature everywhere was roughly the temperature at which rock melts, and he took that to be the Earth’s internal temperature (assuming that the core had not cooled by much since). This model implied that the Earth was nearly homogeneous in its physical properties. Unfortunately, even this ingenious scheme did not fully solve the problem, since the value of the fusion temperature of rock was not known in Kelvin’s time. He was, therefore, forced to adopt an educated guess of seven thousand to ten thousand degrees Fahrenheit for an acceptable range. (Seismic measurements performed in 2007 gave a temperature of about 6,700 degrees Fahrenheit for a region that is about 1,860 miles below the Earth’s surface.)

Putting together all of this information, Kelvin finally computed an age for the Earth’s crust: ninety-eight million years. Estimating the uncertainties in his assumptions and in the data available to him, Kelvin believed that he could state with some confidence that the Earth’s age had to be somewhere between twenty million and four hundred million years.

In many respects, in spite of the insecure assumptions, this was a truly brilliant calculation. Who would have thought that one could actually calculate the age of the Earth? Kelvin took a seemingly insoluble problem and deciphered it. He used sound physical principles both in the formulation of the problem and in his method of calculation, and he augmented those by the best quantitative measurements available at the time (some of which he performed by himself). Compared with his determination, the geologists’ estimates appeared to be nothing more than crude guesses and idle speculation based on poorly understood processes such as erosion and sedimentation.

The number that Kelvin produced—roughly one hundred million years—was broadly consistent with an earlier estimate he had made of the age of the Sun. This was significant, since even some of Kelvin’s contemporaries realized that the strength of his argument about the age of the Earth derived at least part of its credibility from his solar calculation. Kelvin’s basic premise in the paper “On the Age of the Sun’s Heat,” and in a few similar later papers, was not very different from his central thesis in his analysis of the age of the Earth. The key assumption was that the only source of energy that the Sun had at its disposal was the mechanical gravitational energy. This was supposed to be supplied either by the falling-in of meteors, as Kelvin originally thought, but later rejected, or, as Kelvin proposed later and forcefully reiterated in 1887, by the Sun continually contracting, and dissipating its gravitational energy in the form of heat. Since, however, the energy supply was clearly not infinite, and the Sun was unceasingly losing energy by radiation, Kelvin concluded justifiably that the Sun could not remain unchanged indefinitely. To calculate its age, he borrowed elements from theories for the formation of the solar system proposed by the French physicist Pierre-Simon Laplace and the German philosopher Immanuel Kant. He then supplemented those with important insights on the Sun’s potential contraction gained from the work of his contemporary German physicist Hermann von Helmholtz. Weaving all of these ingredients into one coherent picture, Kelvin was able to obtain a rough estimate of the Sun’s age. The last paragraph in Kelvin’s paper reflected his acknowledgement of the many uncertainties involved: