54° 



NA TURE 



{Oct. 31, 1872 



10 >: 100 = 1,000 times grcalcr than an equal radial depth at /. 

 l!ul, in computing tlic dynamic energy de^■eluped liy the shrinlving 

 of the sun, it must l)e borne in mind that a particle at a falls 

 through a distaiice ten times greater than a particle at /. The 

 length of tlie ordinatcs of the curve / /, Fig. 3, representing the 

 ratio of dynamic energy developed at the respective distances 

 from the sun's centre, has been calculated accordingly. A cur- 

 sory examination of Fig. 2 can scarcely fail to lead to the con- 

 clusion that the mass composing the smaller sections of the 

 spherical pyramid towards the centre of the sphere, will be at- 

 tracted liy the larger mass composing the sections towards the 

 circumference. Newton has disposed of this question by a geo- 

 metrical demonstration which, considering the form of the 

 attracting mass, and the extreme complication arising from the 

 varying direction and unequal magnitude of the attracting forces, 

 may be regarded as one of the most elegant of his masterly de- 

 monstrations of important propositions and theorems. It will be 

 evident on rellection that, unless it can be proved that a particle 

 at P is not attracted by any portion of the mass contained within 

 the outer spherical tuperfices / K S and the interior spherical 

 superfices P p, the mass composing the sections near the base of 

 the spherical pyramid will exert the disturbing attraction before 

 alluded to. Our demonstration of the energy produced by the 

 attraction of the matter within the sun, during shrinking, falls to 

 the ground, unless it can be shown that every particle composing 

 the spherical pyramid is in perfect repose as regards the attrac- 

 tion exerted by exterior particles. The great geometer thus 

 establishes that repose : — Let H I K L be a spherical superfices, 

 and P a corpuscle placed within.* Through P let there be 

 drawn to this superfices the two lines UK, IL, intercepting very 

 small arcs II I, A' L ; and because the triangles II P I,Lr K 

 are homogeneous, those arcs will be proportional to the dis- 

 tances ///', L P ; and every particle at /// and KL of the 

 spherical superfices, terminated by right lines passing through 

 P, will be in duplicate ratio of those distances. Therefore the 

 f.jrces of these particles exerted upon the body P are equal be- 

 tween themselves. F'or the forces are as the particles directly, 

 and the squares of the distances inversely. And these two ratios 

 compose the ratio of equality. The attractions, therefore, being 

 made equally towards contrary parts, destroy each other. And, 

 by a like reasoning, all the attractions through the whole spheri- 

 cal superfices are destroyed by contrary attractions. Therefore 

 the body /' will not be anyway impelled by those attractions. 



Referring to Fig. 3, let us recollect that the ordinates of the 

 curve //do not indicate the force exerted by mere attraction. 

 As already stated, their length represents the dynamic energy 

 developed at definite distances between the centre and the cir- 

 cumference of the sphere. The energy actually produced is 

 represented by the superficies op t, wdiile the rectangle op n t 

 represents the energy that would be called forth if the force ex- 

 erted at every point of the axis of the spherical pyramid were 

 the same as that exerted at a a. Our space will not admit of 

 introducing the calculations by which the energy represented by 

 the ordinates of the curve / / have been computed. It will be 

 proper, however, to call attention to the fact that the energy 

 exerted at each of the divisions of the base line ot is definite ; 

 hence the length of the ordinates is exact. Calculations based 

 on the data thus furnished show that the superficies opt is 

 o '2001 5 of the superficies ()/« /. 



We have before slated that the want of homogeneity of the 

 solar mass will not materially aflect the amount of energy de- 

 veloped by the gravitating force during the sun's shrinking. 

 Referring to the several figures, it will be seen that the energy 

 exerted at a point halfway from m, viz., ordinate 5, is o'o625, 

 or yV of that exerted at a a ; and that the energy developed by 

 the mass contained w ithin the spherical pyramid fm 5 amounts 

 to only s'y of that developed by the gravitation of the mass con- 

 tained within the spherical pyramid a m a'. Now the volume of 

 the spherical pyramid/w/ 5 represents that of a sphere the dia- 

 meter of which is »ne half of the sun, while the spherical pyra- 

 mid a m a' represents the volume of the entire solar mass. The 

 energy resulting from the gravitation of the central spherical 

 mass P p being thus only jV of the energy exerted by the spheri- 

 cal mass IKS, it will be perceived that the degree of density of 



* Sir Isaac Newton, in his demonstrations relating to splierical bodies, 

 supposed these to be composed of an inHnitc number of spherical superficies 

 the thickness of which he thus dctines;— " By the superficies of which I 

 here imagine the solids composed. I do not mean superficies purely mathc- 

 matica!, but orbs so C-xtremely thin that their thickness is as nothing ; that 

 orbs of which the sphere will at Last consist, when the 

 rtjs is increased, and their thickness diminished without 



the matter towards the sun's centre will not materially affect 

 the result of our calculations founded on perfect homogeneity. 



Wc may now proceed to ascertain the amount of dynamic 

 energy produced by the assumed shrinking of the axis of the 

 spherical pyramid a m a' . Having already demonstrated that 

 the .said energy will be 0'200I5 of that produced by the gravita- 

 tion of a homogeneous mass, the section of which is one square 

 foot extending from the[ surface to the centre, it only remains to 

 determine the weight of one cubic foot at the surface of the sun. 

 The specific gravity of the solar mass being 85'() pounds per 

 cubic foot, while the sun's attraction is 27'2 times greater than 

 terrestrial attraction, the weight of one cubic foot at the solar 

 surface will be 27'2 x 85'6 = 232S'3 pounds. Multiplying 

 this weight by the sun's radius expressed in feet, we have, 

 232S'3 = 2,250,821,000 = 5,240,633,000,000, which product, 

 multiplied by o'200i5, shows that the gravitating energy of the 

 matter contained in the spherical pyramid, exerted during a 

 longitudinal contraction of one foot, amounts to 1,048,912,000,000 

 foot pounds. Dividing this latter product by the solar energy 

 per minute, already stated, we find that 4355 minutes, = 3^024 

 days will elapse before the energy produced by constant solar 

 radiation equals the gravitating energy exerted during the 

 shrinking of one foot of the solar radius. The length of one 

 year, 365'25 days, being divided by 3'024, we learn that the 

 annual shrinking of the sun's radius amounts to I20'7 feet. The 

 foregoing figures prove that, notwithstanding this apparently 

 great contraction, a period of 1864 years is necessary to di- 

 minish the sun's diameter — . It hardly requires explana- 



10,000 

 tion that this result is reached by dividing the sun's diameter by 

 10,000 times the stated annual shrinking. 



llelmholtz, in accordance with Laplace's remarkable nebular 

 hypothesis, asserts that the continuation of the original con- 

 densation of the matter composing the sun develops an amount 

 of mechanical energy capable of generating sufiicient heat to 

 make good the present solar emission. According to his calcu- 

 tions, the sun's diameter will be reduced rTrcnru in the course of 

 2,000 years. The practical data assumed by the eminent 

 physicist being less accurate than those upon which our calcula- 

 tions are based, the discrepancy regarding time, 2,000 years 

 against 1S64 years, necessary to effect the stated shrinking of the 

 sun's diameter, may be satisfactorily explained. It will be well 

 to observe that the intensity of the radiant heat will not diminish 

 with the diminished size of the sun. On the contrary, for a 

 given area of the solar surface, the dynamic energy produced by 

 a given rate of shrinking will be increased, since the mass remains 

 the same, while the attraction is inversely proportional to the 

 square of the distance fronr the centre. But the rale- will di- 

 minish with the contraction of the sphere ; hence a shrinking of 

 iVth of the sun's diameter, instead of occupying 1,000 X 1S64 

 = 1,864,000 years, will require somewhat more than 2,000,000 

 years. At the end of that period the gravitating energy will 

 continue to develop, as at present, an amount of dynamic energy 

 represented by 312,000 thermal units per minute for each super- 

 ficial foot ; but the radiating surface, i.e., the area of the solar 

 disc, will have diminished in the ratio of 10- to 9°. 



The present maximum temperature produced by scalar radia- 

 tion on the ecliptic when the earth is in aphelion, being 67' 2, 

 while the intensity of radiant heat diminishes as the area of the 

 radiating surface, it follows that, at the end of 2,000,000 years 

 from the present time, the tropical solar intensity will be reduced 



q« X 67'2 

 to 5 — - = 54'4°i unless Prof. Tyndall's opinion is correct, 



that the earth, in common with the other planets, must "creep 

 in, age by age, towards the sun."* But the pace is no doubt so 

 slow that our calculations will not be seriously affected ; hence, 

 applying the foregoing demonstrations to the past, it will be seen 

 that the temperature called forth by solar radiation 2,000,000 

 years ago must have been, owing to the greater diameter of the 



sun at that period, about 



Il3 X 67-2 



= Si" 



within the tropics. 



Nov7 we are justified in assuming that the increased 

 evaporation ot the sea, and the consequent humidity 

 of the atmosphere, modified the stated solar intensity, 

 calling forth the luxuriant flora of past ages, which 

 geology has made us acquainted with. The computed diminu- 

 tion of solar intensity, 67° - 54° = 13°, during the next 

 2,000,000 years will probably be deemed extravagant by those 

 who do laot bear in mind that the computation must be based on 

 * See " Heat at a Mode »f Motion," p. 499. 



