[1] For the apparent brightness of an object falls off as the inverse square of its distance, and the square of 80,000 is approximately equal to 6000 million.

[2] Let C be the centre of the earth, and bCD the diameter through b. Then

BA² = Bb × BD,

where Bb = 16 feet, and BD, which is 16 feet more than the earth’s diameter = 41,900,000 feet. From this we readily calculate that BA = 25,880 feet. This calculation of course neglects the height of the hill AAʹ by comparison with the earth’s diameter.

[3] Here, as throughout the book, we use the French or metric ton of a million grammes or 2204·5 lbs. The English ton of 2240 lbs. is equal to 1·0160 French tons.

[4] The notation 6 × 10²¹ stands for the number formed by a 6 followed by 21 zeros, this shorthand notation being essential, in the interests of brevity, in discussing astronomical numbers. A million is 10⁶, a million million is 10¹² and so on.

A similar notation is needed to express very small numbers. The expression 10⁻²¹ is written for 1/10²¹ and so on. Thus 6 × 10⁻⁶ stands for 6/1,000,000 or 0·000006.

[5] The simplest definition of an ellipse is that it is the curve drawn by a moving point P which moves in such a way that the sum of its distances PS, PT from two fixed points S, T remains always the same. In practice we can most easily draw an ellipse by slipping an endless string SPTS round two drawing pins S, T stuck into a drawing board. Stretch the string tight with a pencil at P, and on letting the pencil move round, keeping the string always tight, we shall draw an ellipse. If the pins S, T in the drawing board are placed near to one another the curve described by the pencil P is nearly circular. The ratio of the distance ST to the length of the remainder of the string SP + PT is called the “eccentricity” of the ellipse; it is necessarily less than unity, because two sides of a triangle are together greater than the third side.

In the limiting case in which the eccentricity is made zero, the ellipse becomes a circle. If the eccentricity is nearly as large as unity, the ellipse is very elongated. All the different shapes of ellipses are obtained by letting the eccentricity change from 0 to 1, and these represent all the different shapes of orbit that a small body can describe around a heavy gravitating mass. The points S, T are called the foci of the ellipse, and the big attracting body always occupies one or other of the two foci of the ellipse.

[6] What he actually observes is the “projection” of the orbit on the sky, but it is a well-known theorem of geometry that the projection of an ellipse is always an ellipse.

[7] For instance, Cepheids whose light fluctuates in a period of 40 hours have approximately a luminosity 250 times that of the sun, and so are of 8 × 10²⁹ candle-power; a period of ten days indicates a luminosity 1600 times that of the sun, or a candle-power of 5·17 × 10³⁰, and so on. If a star in a distant astronomical object is observed to fluctuate with a period of ten days, and the quality of its fluctuations shew it to be a Cepheid variable, we know that its actual candle-power must be 5·17 × 10³⁰. Its apparent brightness is observed to be that of a star of, say, magnitude 16, which, stripped of technicalities, means that we receive as much light from it as from a single candle at a distance of 570 miles. The difference between one candle and 5·17 × 10³⁰ candles accordingly corresponds to the difference between 570 miles and the distance of the object in question, whence, since light falls off as the inverse square of the distance, we calculate that the distance of the object must be

√5·17 × 10³⁰ × 570 miles

or about 220,000 light-years.

[8] This is expressed in the mathematical formula ½mv² for the energy of motion of a body of weight m moving with a speed v. If m is measured in grammes, and v in centimetres per second, the energy of motion of the body is said to be ½mv² “ergs.” Thus an “erg” is the energy of motion of a body of 2 grammes weight (so that ½m= 1) moving with a speed of one centimetre a second. As an example, the energy of an express train of 300 tons’ weight (3 × 10⁸ gms.) moving at 60 miles an hour (2682 cms. a second) is 1079 × 10¹⁴ ergs; a cannon-ball or shell weighing a ton and moving at 1520 feet a second has precisely the same energy.

[9] These were called δ-rays by Bumstead.

[10] The wave-length in a system of ripples is the distance from the crest of one ripple to that of the next, and the term may be applied to all phenomena of an undulatory nature.

[11] The reader whose interest is limited to astronomy may prefer to proceed at once to Chapter III.

[12] To be precise, if v is the frequency of the radiation, its quantum of energy is hv, where h is a universal constant of nature, known as Planck’s constant. This constant is of the physical nature of energy multiplied by time; its numerical value is:

6·55 × 10⁻²⁷ ergs × seconds.

[13] In the form of an equation:

E₁ - E₂ = hν,

where E₁, E₂ are the energies of the material system before and after the change, ν is the frequency of the radiation, and h is Planck’s constant already specified.

[14] The mathematician will readily see the reason for this rule, which is, in brief, as follows: the energy needed to separate two electric charges + e and - e, at a distance r apart, is e²/r, and the energy needed to re-arrange or break up a structure of electrons and protons of linear dimensions r will generally be comparable with this. If λ is the wave-length of the requisite radiation, the energy made available by the absorption of this radiation is the quantum hC/λ. Combining this with the circumstance that the value of h is very approximately

we find that the requisite wave-length of radiation is about 860 times the dimensions of the structure to be broken up.

[15] The wave-length λ of the radiation and the associated temperature T (measured in Centigrade degrees absolute) are connected through the well-known relation:

λT = 0·2855 cm. degree.

[16] On combining the relation just given between T and λ with that implied in the rough law of the “860-limit,” we find that a structure whose dimensions are r cms. will begin to be broken up by temperature-radiation when the temperature first approaches ¹/₃₀₀₀ r degrees.

[17] If we suppose that re-arrangements of an electric structure can also be effected by bombarding it with material particles, the temperature at which bombardment by electrons, nuclei, or molecules first becomes effective is about the same as that at which radiation of the effective wave-length would first begin to be appreciable; the two processes begin at approximately the same temperature.

[18] In discussing the earth’s age, I have borrowed extensively from Professor Holmes’ book, The Age of the Earth.

[19] The eccentricity of orbit e is distributed in such a way that all values of e² from e² = 0 to e² = 1 are equally probable.

[20] This is near enough, but not absolutely accurate. Exact mathematical analysis shews that the weight of the minimum condensation M is given by

where C, γ, ρ, κ are the molecular velocity, gravitation constant, initial density, and ratio of specific heats, whereas the weight from which molecules moving with velocity C just fail to escape is given by

With κ = 1⅔ the minimum weight of condensation is 9·7 times the weight which is just adequate to retain the molecules.

[21] The following estimates have already been mentioned:

[22] Although the details are unimportant, the actual course of events would be that the earth would begin to describe an elliptic instead of a circular orbit about the sun, the earth’s average distance being greater than now.

[23] This is shewn in fig. 15, the area of the 3000 degree curve being only a sixteenth of the area of the 6000 degree curve.

[24] The large round images of stars shewn in the frontispiece result merely from over-exposure, and have nothing to do with the sizes of the stars.

[25] For purely practical reasons the height is not taken proportional to the luminosity but to its logarithm; without some such device as this it would be impossible to represent the range of more than 1,000,000 to 1 in the observed luminosities of red stars.

[26] This provides a further objection to Russell’s hypothesis, which, to avoid confusion, was not mentioned on p. 295.