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The draftsmanship is so accurate that it is easy to deduce from it beautiful theorems, which would be very laborious to work out by the calculus. I submit the geometrical masterpieces to my chief inspector, who is himself said to be smitten with geometry. I modestly describe the method of construction, I call his attention to the fine deductions which the drawing enables one to make.
The savage who executes unerringly the exact throw which brings down his game, or his enemy, in the manner most suited to his purpose, under the operation of all the conditions necessarily involved, the weight and form of the weapon, the direction and distance of the object, the action of the wind, etc., owes this power to a long series of previous experiments, the results of which he certainly never framed into any verbal theorems or rules.
Now, the reasons of a maxim of policy, or of any other rule of art, can be no other than the theorems of the corresponding science. The relation in which rules of art stand to doctrines of science may be thus characterized. The art proposes to itself an end to be attained, defines the end, and hands it over to the science.
In geometry, instead of the tedious apparatus of fallacious proofs for obvious truisms which constitutes the beginning of Euclid, the learner should be allowed at first to assume the truth of everything obvious, and should be instructed in the demonstrations of theorems which are at once startling and easily verifiable by actual drawing, such as those in which it is shown that three or more lines meet in a point.
But he seems to me to be one practical in this sense, that his theorems will shape life for him, directly; that he will always seek, as a matter of course, the effective equivalent to the line of being which shall be the proper continuation of his line of thinking.
To give a colourable aspect to their hypothesis, the witnesses affirm that Lévi was a high-grade Mason. He was nothing of the kind; he affirms most distinctly in his "History of Magic," that for any knowledge which he possessed about the mysteries of the fraternity, he owed his initiation only to God and to his individual studies. Secondly, the practice of ceremonial magic, which is what the witnesses understand by theurgy, is a practice condemned by Lévi, except as an isolated experiment to fortify intellectual conviction as to the truth of magical theorems. He attempted it for this purpose in the spring of the year 1854, and having satisfied himself as to the fact, he did not renew it. Thirdly, the philosophy of Éliphas Lévi is in direct contrast to Manichæan doctrine; it cannot be explained by dualism, but must be explained by its opposite, namely, triplicity in unity. He shows that "the unintelligent disciples of Zoroaster have divided the duad without referring it to unity, thus separating the pillars of the temple, and seeking to halve God" (Dogme, p. 129, 2nd edition). Is that a Manichæan doctrine? Again: "If you conceive the Absolute as two, you must immediately conceive it as three to recover the unity principle" (Ibid.). Once more: "Divinity, one in its essence, has two fundamental conditions of being necessity and liberty" (Ibid., p. 127). And yet again: "If God were one only, He would never be Creator nor Father. If He were two, there would be antagonism or division in the infinite, and this would be severance or death for every possible existence; He is therefore three for the creation by Himself, and in His image of the infinite multitude of beings and numbers. Thus He is really one in Himself and triple in our conception, by which we also behold Him triple in Himself and one in our intelligence and in our love. This is a mystery for the faithful and a logical necessity for the initiate of the absolute and true sciences" (Ibid., p. 138). And the witnesses of Lucifer have the effrontery to represent Lévi as a dualist! I will not discredit their understanding by supposing that they could misread so plain a principle, nor dissemble my full conviction that they acted with intentional bad faith. Fourthly, Éliphas Lévi regarded Lucifer as a conception of transcendental mythology, and the devil as an impossible fiction, or an inverted and blasphemous conception of God divinity
Thus not only the development of applied geometry but the reduction of other conceptions to a rigorous mathematical form was indefinitely postponed. There is, however, one science which admitted of the immediate application of the theorems of geometry, and which did not require the application of the experimental method.
Matters went steadily worse with Paul at school. In the itch to let his instructors know how heartily he despised them and their homilies, and how thoroughly he was appreciated elsewhere, he mentioned once or twice that he had no time to fool with theorems; adding with a twitch of the eyebrows and a touch of that nervous bravado which so perplexed them that he was helping the people down at the stock company; they were old friends of his.
By means of the Arithmetical Triangle, an invention equally ingenious and original, he succeeded in solving a number of theorems which it would have been difficult to demonstrate in any other way, and in finding the coefficients of different terms of a binomial raised to an even and positive power.
His business as he saw it was with the material facts of life, or, rather, with those third and fourth degree theorems and syllogisms which control material things and so represent wealth. He was here to deal with the great general needs of the Middle West to seize upon, if he might, certain well-springs of wealth and power and rise to recognized authority.
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