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deal with abstract and necessary forms of Human thought. They disclose to 

 us an infinity of ratios, or relations, subsisting between the various ideas of 

 number and magnitude with which they deal ; comprising the properties of 

 geometrical figures, plane and solid, triangles, squares, circles, ellipses, prisms, 

 cylinders, cones, spheres, etc. Now, on coming to the examination of external 

 Nature, Man finds to his amazement that Nature '• geometrizes " in all her 

 departments. There is a definite apportionment of Space and Time, there are 

 definite relations of Number and Magnitude, underlying, as it were, all Natural 

 operations. The geometrical webs spun by man in his own brain, with ideal 

 lines, turn out to be the ground-plans of Nature herself. The planets, to take 

 a familiar instance, move round the sun in elliptical orbits, having the sun for 

 a common focirs. Their speed in different parts of their orbits is governed by 

 a law capable of precise geometiical expression ; for every planet moves in 

 such a way that the line drawn from it to the sun sweeps over equal areas in 

 equal times. An exact arithmetical relation subsists between the periods of 

 revolution of the several planets and their respective distances from the Sun ; 

 the squares of the period being proportional to the cubes of the mean distances. 

 Again Bode's law discloses a rather remarkable numerical harmony in the 

 progression of the distances of the planets from the Sun. 



The general regularity of this series (a series in duple progression), was 

 early observed : but the rule seemed to be broken in the case of the wide 

 interval between Mars and Jupiter, where a member of the system seemed 

 wanting. Bode argued that a planet must exist to fill up this gap ; and 

 towards the close of the last century there began a search for it. This has 

 resulted in the discovery of a whole family of comparatively minute bodies, 

 which may have been fragments of a larger one. Collectively, at all events, 

 these planetoids fulfil the expectation of Science, for they revolve in orbits at 

 a mean distance from the sun almost exactly corresponding with that indicated 

 by Bode's law, as the proper distance of the missing member of the system.* 



Chemistry gives limits of the mathematical groundwork of Nature as 

 distinct, almost, as those conveyed by the queenly science of Astronomy. 

 The supposed primitive elements of bodies, Oxygen, Hydrogen, Nitrogen, 

 Carbon, and the rest, in whatever quantities they are mixed, combine with one 

 another only in constant numerical proportions. When, as is commonly the 

 case, one substance combines with another in several different proportions, the 

 higher proportions are multiples of the lowest. This gives to the formulae of 

 Chemistry the very aspect of an algebraic series. Even the laws of musical 

 concord depend upon the ratio subsisting between the numbers, in a given time, 

 of the vibrations which produce the notes. In the simple chord of three notes, 

 or harmonic triad, the Dominant performs four vibrations, whilst the Third 

 performs five, and the Fifth six ; and the superior, because more readily perceived, 

 harmony of the combination is dependent upon the simplicity of this ratio. 

 I believe there is little doubt that harmony of colour dej)ends on a similar 



* Even more striking than these instances, is the fact, that the law of gravity itself 

 may be regarded as the simple expression of an a priori truth dependent upon the abstract 

 conception of Force, and on the geometrical relation subsisting between the superficial 

 areas of spheres of different magnitudes. Suppose a force emanating from the centre of 

 several concentric spheres, and diffusing itself through space. Taking it for an axiom, 

 that Force is never lost, the supposed Force will become attenuated in proportion to the 

 distance from the seat of power (the common centre), but will remain, in sum, undi- 

 minished. The sum of the force exerted on the surface of each sphere will then be the 

 same. But these surfaces are in direct proportion to the squares of the radii of the 

 spheres. The force, therefore, on any given part of any of these surfaces must, in its 

 intensity, be in the inverse ratio of the radius— i.e., inversely as the distance. The 

 undulations of light, heat, and sound, follow the same law. The first law of motion is 

 also deducible, a priori, from the abstract idea of Force. 



