46. Art is the collection of rules for doing any thing well; and is the more perfect in proportion as each rule embraces a greater number of cases, and consequently as the number of these rules is smaller. Doubtless, buildings that were solid, well proportioned, and adapted to the purpose for which they were destined, had been constructed before the rules of architecture were reduced to formulas; but the great progress of intelligence in the construction of buildings consisted in ascertaining what there was common to all well-built houses, in determining the cause of beauty and of solidity, in themselves considered, by passing from the individual to the universal, that is, by forming general ideas of beauty and solidity applicable to an indefinite number of particular cases, by simplifying.
47. The same may be said of all other liberal and mechanical arts: the progress of intelligence in all of them consists in reducing multiplicity to unity, and including the greatest possible number of applications in the least possible number of ideas. This is why lovers of literature and the fine arts labor to discover an idea of beauty in general, in order to attain a type applicable to all literary and artistic objects. It is also obvious that those engaged in mechanical arts always endeavor to govern their proceedings by a few rules, and he is held to be the most skilful who succeeds in combining the greatest variety of results with the greatest simplicity of means, by making that, which others connect with many ideas, depend upon one idea alone. When we see a machine produce wonderful effects by a very simple process, we praise the artificer not less for the means than for the end: this we say, is grand, and the simplicity with which it works is the most astonishing.
48. Let us apply this doctrine to the natural and exact sciences.
The merit of our actual system of numeration consists in including the expression of all numbers in a single idea, making the value of each figure ten times that to the right, and filling all intervals with zeros. The expression of infinite numbers is reduced to the simplicity of a single rule based upon a single idea; the relation of position with a tenfold value. Logarithms have enabled arithmetic to make a great advance by diminishing the number of its fundamental operations, since, with them it reduces multiplication and division to addition and subtraction. Algebra is only the generalization of arithmetical expressions and operations, their simplification. The application of algebra to geometry is the generalization of geometrical expressions; formulas of lines, figures, bodies, only the expression of their universal idea. In this idea as in a type, geometry preserves its first and generative idea, and it requires only the simplest applications in order to form an exact calculation of all lines belonging to the same class, which can possibly be met with in practice. In the simple expression dz/dx = A, called the differential coefficient, is contained the whole idea of infinitesimal calculus. It originated in geometrical considerations, but so soon as its universality was conceived, it poured a flood of light upon every branch of mathematical and natural science, and led to the discovery of a new world, whose confines are still unknown. The prodigious fecundity of this calculus emanates from its simplicity, its prompt generalization of both algebra and geometry, and its uniting them in a single point which is the relation of the limits of the differentials of any function.
49. It is to this unity of idea that the human intellect in its ambition aspires, and once obtained, it proves the cause of great progress. The glory of the greatest geniuses is that they discovered it: the advance of science has consisted in profiting by it. Vieta explained and applied the principle of the general expression of arithmetical quantities; Descartes extended this to geometrical quantities. Newton established the principle of universal gravitation; and he, at the same time with Leibnitz, invented the infinitesimal calculus; and the exact and natural sciences march, by the light of a vast flambeau, with gigantic strides along paths never before trodden. And all this because intelligence has approached unity, and become possessed of a generative idea, involving infinite other ideas.
50. It is worthy of remark, that as we advance in science, we meet numerous points of contact, close relations, which no one at first sight would have suspected. Ancient mathematicians discussed the conic sections, but were far from imagining that the idea of the ellipse could be the basis of a system of astronomy: the foci to them were simple points, the curve a line, and the relations of both the object of combinations at once profitless and without application. Ages pass away, and these foci are the sun, the curve the orbit of planets. The lines on the geometrician's table represented a world!
The intimate connection of mathematical and natural science cannot be questioned; and who shall say to what extent both are connected with ontological, psychological, theological, and moral science? The extended scale over which beings are distributed may at first sight seem to be an assemblage of unconnected objects, but seen with the eyes of science, it is perceived to be a delicately worked chain, whose links present, as we advance, greater beauty and perfection. We see the different realms of nature united by close relations: the sciences, of which they are the objects, mutually borrow each other's light, and enter on each other's territory. The complication of objects among themselves involves this complication of science; and the unity of the laws imposed upon different orders of beings makes all sciences approach, and tend to form, one only science. If it were given us to see the identity of their origin, the unity of the end and the simplicity of the means, we should come into the possession of the true transcendental science, the only science which involves all others, or more correctly speaking, the only idea in which every thing is represented as it is, and every thing seen without any necessity of combination, or effort of any kind, just as a magnificent landscape, its outlines, form, and colors are pictured on a perfectly clear mirror. In the meantime, we must rest satisfied with shadows of reality, and must see in the instinctive tendency of our understanding to simplify, to reduce every thing or make it approach as much as possible to unity, the announcement, the sign of this single science, this intuition of the one infinite idea; just as in the desire for happiness which agitates our heart, the thirst after enjoyment which torments us we discover a proof that all is not ended here below, and that our soul has been created for the possession of a good not to be attained in this mortal life.
51. If we compare men with men, and pay attention to the character of genius, the most elevated point of human intelligence, we shall see the truth of what has been said of the scale of human beings, and the progress of science. Men of true genius are distinguished by the unity and extent of their conceptions. If they treat a difficult and complicated question, they simplify it, consider it from a high point of view, and determine one general idea which sheds light upon all the others. If they have a difficulty to solve, they show the root of the error, and with a word dispel all the illusion of sophistry. If they use synthesis, they first establish the principle which is to serve as its basis, and with one dash trace the road to be followed in order to reach the wished-for result. If they make use of analysis, they strike in its secret resort the point where decomposition is to commence, they at once open the object, and reveal to us its most obscure mysteries. If there is question of a discovery, while others are seeking here and there, they strike the ground with their foot, and exclaim, "the treasure is here." They make no long arguments, nor evasions; their thoughts are few but pregnant; their words are not many, but in each of them is set a pearl of inestimable value.
52. No doubt there is in the intellectual order a single truth from which all other truths emanate, one idea which includes all other ideas. This philosophy teaches, and the efforts, the natural and instinctive tendencies of every intelligence, toiling after simplicity and unity, show it: such also is the dictate of common sense, which considers that thought the highest and most noble which is the most comprehensive and the most simple.(4)
CHAPTER V.
TRANSCENDENTAL SCIENCE IN THE HUMAN INTELLECTUAL ORDER CANNOT EMANATE FROM THE SENSES.
53. In the human intellectual order, such as it is in this life, there is no one truth from which all others flow: philosophers have sought one in vain; they have