substances, like stones and metals, which make up the planets. These curious views of Descartes are interesting chiefly as forming a connecting link between the atomic speculations of the Greeks and the more highly developed theories of

modern times.

After Priestley's discovery of oxygen in 1772, and Lavoisier's demonstration some fifteen years later that fire is not an element, but only a process, a combination of the elements of oxygen and carbon, thus liberating molecular and atomic energy, the light of the Sun and stars doubtless appeared in a new aspect. Instead of being made up of the element fire, the stars were now considered to be burning bodies.

Yet all the various attempts to explain the light and heat of the stars by chemical processes largely failed; and in 1854 Helmholtz showed that a great and steady supply of energy becomes available from the gravitational potential of the Sun's mass, converted into light and heat by slow shrinkage and subsidence of the particles toward the centre.

To make it quite clear how this takes place, we need only recall Joule's experiments on the mechanical equivalent of heat. About sixty years ago this eminent British physicist showed experimentally that if a mass with a weight of one pound be allowed to fall through a space of 772 feet, the heat given up by the falling body would be adequate to raise the temperature of one pound of water one degree Fahrenheit. Larger bodies would produce more heat in proportion to their masses; and where the force of gravity is larger than on the Earth, as in the Sun, this would give still more heat in proportion to the intensity of gravity. Now at the surface of the Sun the force of gravity is about 28 times what it is upon the Earth; and moreover, the Sun's mass is about 332,750 times that of the Earth. If, therefore, the Sun's force of gravity is so much larger, and it has a so much larger quantity of matter to fall under the action of this force, it follows that the heat devel

oped in the condensation of the Sun must be enormous. To calculate the exact amount of heat developed, we have to make use of the higher mathematics, and also know the law of density within the Sun's globe, which we shall discuss more fully hereafter.

On the supposition that the Sun is homogeneous, or of uniform density throughout, and the heat and light radiated away as fast as produced, a contraction in the radius of 110 feet per annum was found by Helmholtz to be adequate to furnish our enormous supply of light and heat.1 We shall see in the course of this paper that this theory of Helmholtz is only the beginning of our present theories of the Sun; yet it has the great advantage over the old theories of assigning a true cause based on established physical laws, and therefore will remain of interest throughout all time.

The work of Helmholtz thus marks an epoch in our theories of the Sun, and has been the starting-point of nearly all subsequent researches on the subject. But it can hardly be said that the theory of Helmholtz was more fundamental than that of Lane; who treated of the Sun's constitution on the hypothesis that it is a sphere of gas kept in equilibrium under the temperature, pressure, and attraction of its parts. For Helmholtz had only considered the gravitational condensation of a homogeneous Sun of given size and mean density, without inquiring whether it was solid, liquid, or gaseous. He supposed it to have formed according to the nebular hypothesis of Laplace, and therefore, no doubt, assumed that it was originally a gaseous nebula, of which the high temperature might have arisen from the falling together of cold matter, in accordance with Joule's experiments. Lane took up the consideration of the Sun as it is to-day, and worked out some of the most important laws for its internal constitution, showing that the mass must be

1 According to the author's researches based on the Monatomic Theory, the actual shrinkage in the sun's radius is 216 feet per annum.

essentially gaseous throughout, although already of considerable density.

Jonathan Homer Lane was a native of western New York, for many years connected with the Patent Office and Ų. S. Coast and Geodetic Survey in Washington, and a member of the National Academy of Sciences. He was a man of retiring disposition, and, although he did no vast amount of scientific work, what he did was of high quality, and bears unmistakable marks of genius. His paper on the Sun is probably his most famous effort, and has since become classic and justly celebrated. Frequently cited by astronomers of other nations, it is perhaps the most important single contribution since that of Helmholtz in 1854. Lane's paper "On the Theoretical Temperature of the Sun, under the Hypothesis of a Gaseous Mass maintaining the volume by its internal Heat, and depending on the Laws of Gases as known to Terrestrial Experiment," was read to the National Academy of Sciences at the Washington meeting of April 13-16, 1869, and published in the American Journal of Science for July, 1870. This is the famous paper so much quoted by Lord Kelvin, Ball, Newcomb, Perry, and others, who have discussed the mathematical theory of the Sun's heat.

Lane describes the inception of his paper as follows:

"Some years ago the question occurred to me, in connection with this theory of Helmholtz, whether the entire mass of the sun might not be a mixture of transparent gases, and whether Herschel's clouds might not arise from the precipitation of some of these gases, say carbon, near the surface, with their revaporization when fallen or carried into the hotter subjacent layers of atmosphere beneath; the circulation necessary for the play of this Espian theory being of course maintained by the constant disturbance of equilibrium due to the loss of heat by radiation from the precipitated clouds. Professor Espy's theory of storms I first became acquainted with more than twenty years ago from

lectures delivered by himself; and, original as I suppose it to be, and well supported as it is in the phenomena of terrestrial meteorology, I have long thought that Professor Espy's labors deserve a more general recognition than they have received abroad. It is not surprising, therefore, in a time when the constitution of the sun was exciting so much discussion, that the above suggestion should have occurred to myself before I became aware of the very similar, and in the main identical, views of Professor Faye, put forth in the Comptes Rendus. I sought to determine how far such a supposed constitution of the sun could be made to connect with the laws of gases as known to us in terrestrial experiments at common temperatures."

Although Lane's treatment of the Sun's internal constitution was considered highly satisfactory, his mathematical processes were so difficult that very few later investigators have ever worked out his results independently. The subject of the Sun's internal condition was next treated by the German physicist Ritter, of Aixla-Chapelle, in 1878, and a few years later by Lord Kelvin. In 1899 this problem was also treated by Professor John Perry of London, who followed the same general methods as Lane, Ritter, and Kelvin.

An outline of these researches, and of the considerable extension very recently made of them by the writer, is all that would be of interest to the general reader. Before taking up the details of this treatment, however, it is necessary to remark that, while in these calculations full account is taken of the energy of gravitation arising from the mutual approach of the particles under gravity, no attention is given to the energy arising from such substances as radium. At present it is not known whether radium exists in the stars, but, as it exists in the Earth, it has been held that it must also exist in the Sun, or will develop there some time in the future when our star cools down to a stage corresponding to that now occupied by the

Earth. We shall recur to this subject again toward the end of this paper.

Assuming that the only energy given out by a condensing body is that derived from the gravitational attraction of the particles, Helmholtz in 1854 showed that the total heat produced up to the present time in the condensation of the Sun would raise the temperature of an equal mass of water about 27,000,000° Centigrade. As Pouillet found by experiment that the annual radiation of the Sun was adequate to cool an equal mass of water 1.25° Centigrade, it followed that the total duration of the Sun's activity at this uniform rate of radiation could not exceed some twenty million years, which very markedly curtailed the past duration of the Earth as inferred by geologists from the study of phenomena of the Earth's surface.

Helmholtz's theory was somewhat defective, in assuming the density of the Sun's globe to be uniform throughout; but as a first approximation to the laws of nature it met all requirements, and, indeed, marked an important epoch in the history of scientific thought during the nineteenth century.

In Lane's paper the conclusion was reached that the Sun is really quite heterogeneous, the central density being some 20 times the mean. This result was based upon the hypothesis that the solar gases are like oxygen, hydrogen, nitrogen, and common air, in which the ratio of the specific heat of the matter under constant pressure to that under constant volume is k=1.4. The value of k always plays an important part in the theory of the Sun; for upon this physical constant depend the laws of internal density, and therefore, also, the total heat developed up to the present time, as well as the pressure and temperature throughout the Sun's globe.

II

The writer has recently carried out the most elaborate investigation of the mathematical theory of the Sun yet attempted,

and published the results in the Astronomische Nachrichten, noumber 4053. On carefully examining the work of Lane, Kelvin, and Ritter, it was found that they could all be reconciled quite perfectly among themselves by correcting a misconception in the paper of Lane.

This was to the effect that the Sun's atmosphere extends above the photosphere by one twenty-second part of the radius. Though it is now known that this assumption is not justifiable, the misconception misled Lord Kelvin, and caused him and other eminent writers to conclude that the central density of the Sun, conceived as made up of biatomic gases, should be about 20 times the mean density, whereas it should be a little over 23 times the mean. By a different process Lord Kelvin concluded that the central density should be 22.5 times the mean density, while from certain equations of the celebrated French mathematician, Poisson, Ritter found 23 to be the proper number.

When it was found by the writer's recent researches that Lane's theory, correctly interpreted, made the central density about 23.4 times the mean, instead of 20 times, as given in the published paper of 1869, it was seen that all three determinations of the internal laws of the Sun's density were essentially in perfect agreement. The rigor of the gaseous theory of the Sun's constitution was thus confirmed by the accordant results reached by three independent processes, and there can be no doubt of the accuracy of the final value.

These investigations, however, in which the ratio of the specific heat of the gas under constant pressure to that under constant volume is k=1.4, as in biatomic gases like oxygen, nitrogen, hydrogen, air, do not correspond to the conditions in nature, where the temperature is enormously high, and we shall consider more particularly the case in which k=13. This corresponds to a monatomic gas, or a gas in which the molecules are identical with the atoms, and may be

viewed as single spheres without mutual connections of any kind. Ordinary gases, such as oxygen, hydrogen, nitrogen, have two atoms in a molecule, probably joined together like the two ends of a dumb-bell, while the more complex gases have molecules made up of many atoms grouped together in various ways.

Now when the molecules are very complex, made up of many atoms variously arranged, the group thus formed frequently becomes unstable. The parts are always in rapid motion, and a molecule may be likened to a political convention, which is made up of many individuals, and has correspondingly unsteady qualities.

It is found by experiment that all complex gases are decomposable at some temperature not enormously high. Vapor of water and ammonia are dissociated into their constituent atoms at temperatures less than 1000° Centigrade, and probably all the chemical bodies we know of would be dissociated at temperatures less than 10,000° Centigrade, or 18,000° Fahrenheit. At all higher temperatures chemical compounds probably cease to exist, and the molecules of the substances are reduced to the state of single atoms, and hence called monatomic.

Such we conceive to be the state of the matter in the Sun. For it is shown by observation and calculation that the fixed stars and the Sun have internal temperatures of many millions of degrees, while at their surface the temperatures will seldom fall short of 10,000° Centigrade. We may, therefore, take the whole interior of the Sun and stars as monatomic gas; and suppose that even at their surfaces few compounds can form, so that, in general, the body of stars composing the visible universe are flaming globes of monatomic gas, in which all the elements are reduced to their simplest form of single atoms.

No doubt our Sun is a globe of this kind, but it has usually been treated as made up of compound gases, like air, hydrogen, oxygen, nitrogen. What, then,

is the arrangement of its internal density when the gases are monatomic?

Lane began to consider this question as far back as 1869, treating the Sun's globe as possibly made up throughout of monatomic gas; and the mathematical methods employed by him have recently been much extended and improved by the writer of this paper. These processes depend on the development of certain series based on methods of the higher mathematics, of which an account here would be out of place. Suffice it to say that the investigation as thus carried out involved the calculation of numbers running up into the hundreds of sextillions, that is, numbers expressed by twenty-four places of figures. These numbers are so stupendous as to be almost unmanageable, and the work had to be done by the oldfashioned direct processes, without the use of logarithms, which are no longer available. This vastly increased the labor of calculation, and also the liability to error, so that all the work had to be repeated three or four times to insure accuracy in the final result. At length the process was made sufficiently accurate, and led to some of the most beautiful results yet attained in any branch of physical science, because apparently applicable to the great body of the fixed stars.

One of these results of great interest is that the central density in a star made up of layers of monatomic gas is exactly six times the mean density. This appears to be a general law of nature. In the case of our Sun, for example, the mean density is 1.4 times that of water; and the density at the centre thus becomes 8.40; which exceeds the density of steel (7.816) and even brass (8.383), and proves to be practically midway between that and German silver (8.432).

An examination of the table on page 769 shows the following facts:

1. The outer layers of the Sun are of the same order of density as our atmosphere, becoming only 153 times the density of air one tenth of the way to the centre, where the pressure is 21,636,565

atmospheres, or 7 times greater than it is at the centre of the earth.

2. The rise of pressure and temperature downward is very rapid. At the centre the pressure is over 11,215,000,000 atmospheres, equivalent to that exerted by

a vertical column of quicksilver about one tenth as long as from the earth to the sun if all parts of the column were under the uniform acceleration of terrestrial mean gravity.

3. The temperature at the centre of the

Sun is about 50,000,000 degrees Centigrade.

4. In the outer layers of the Sun the density rises steadily, the temperature somewhat more rapidly, and the pressure most rapidly of all. The result is that at a moderate depth the pressure becomes so great that circulation under this great strain on the atoms is impossible, on account of the friction of the fluid against itself. Currents observed near the surface of the Sun, therefore, do not extend to any considerable depth, and the matter in the Sun's interior is always kept highly rigid from pressure.

Heretofore astronomers have very generally supposed that the circulation extended throughout the Sun's body.

We shall first examine the effects of this arrangement of the density on the total amount of heat developed in the VOL. 97 - NO. 6

condensation of the Sun. It will be seen from what is said above that when k=1.4, as imagined by Lane, Ritter, Lord Kelvin, and Perry, the central density is 23 times the average for the whole sphere, but when k=13, as in gases reduced to the monatomic state by intense heat, the central density is only six times the mean density. Now, in the theory of the Sun's heat considered by Helmholtz, the density was taken to be uniform throughout. As a heterogeneous Sun can be imagined to result from a homogeneous one by the descent of many of the particles toward the centre, so as to increase the density in that region, we see that when the particles have fallen inward in a certain way the arrangement corresponds to the monatomic sphere, and when still more of them have fallen downward, and nearer the centre, the arrangement corresponds to