On Friday, 13 August 2021 at 20:26:10 UTC+3, Eram semper recta wrote:
> I've spoken at length about Euler's main Blunder S = Lim S. Now I focus on the contradictory contents of his Elements of Algibberish.
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> Euler said it! Not me! From his Elements of Al-gibberish , the following:
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> 82. To express this idea, according to the sense of it above-mentioned, we make use of the sign ∞, which consequently indicates a number infinitely great; and we may therefore say that this fraction 1/∞ is in reality nothing; because a fraction cannot be reduced to nothing, until the denominator has been increased to infinity.
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> 83. It is the more necessary to pay attention to this idea of infinity, as it is derived from the first elements of our knowledge, and as it will be of the greatest importance in the following part of this treatise.
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> We may here deduce from it a few consequences that are extremely curious, and worthy of attention. The fraction 1/∞ represents the quotient resulting from the division of the dividend 1 by the divisor ∞. Now, we know that if we divide the dividend 1 by the quotient 1/∞, which is equal to nothing, we obtain again the divisor ∞: hence we acquire a new idea of infinity; and learn that it arises from the division of 1 by 0; so that we are thence authorised in saying that 1 divided by 0 expresses a number infinitely great, or ∞.
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> 84. It may be necessary also, in this place, to correct the mistake of those who assert that a number infinitely great is not susceptible of increase. This opinion is inconsistent with the just principles which we have laid down; for 10 signifying a number infinitely great, and 20 being incontestably the double of 10, it is evident that a number, though infinitely great, may still become twice, thrice, or any number of times greater.
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> The above claims are wild compared to the first few claims in his book which if there wasn't so much garbage in it, might still have been one of the best algebra text books ever published:
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> Chapter I – Of Mathematics in general
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> 1. Whatever is capable of increase or diminution is called magnitude, or quantity. A sum of money therefore is a quantity, since we may increase it or diminish it. It is the same with a weight, and other things of this nature.
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> JG: In other words, a magnitude is simply the concept of size, dimension or extent.
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> 2. From this definition, it is evident that the different kinds of magnitude must be so various, as to render it difficult to enumerate them: and this is the origin of the different branches of the Mathematics, each being employed on a particular kind of magnitude.
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> * Mathematics, in general, is the science of quantity; or, the science which investigates the means of measuring quantity. *
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> JG: Euler almost got this one right even though the local idiot Malum does not agree:
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> JG: Mathematics is the science of measure and quantity.
> JG: What Euler appears to have missed is that in order to enumerate anything, the concept of number and all its machinery must already be in place.
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> 3. Now, we cannot measure or determine any quantity, except by considering some other quantity of the same kind as known, and pointing out their mutual relation. If it were proposed, for example, to determine the quantity of a sum of money, we should take some known piece of money, as a louis, a crown, a ducat, or some other coin, and show how many of these pieces are contained in the given sum. In the same manner, if it were proposed to determine the quantity of a weight, we should take a certain known weight; for example, a pound, an ounce, etc. and then show how many times one of these weights is contained in that which we are endeavouring to ascertain. If we wished to measure any length or extension, we should make use of some known length, such as a foot.
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> JG: In other words, we need a ratio which is the first kind of (qualitative) measure. In my free article, I explain how all this is done:
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> https://drive.google.com/file/d/1hasWyQCZyRN3RkdvIB6bnGIVV2Rabz8w
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> 4. So that the determination, or the measure of magnitude of all kinds, is reduced to this: fix at pleasure upon any one known magnitude of the same species with that which is to be determined, and consider it as the measure or unit; then, determine the proportion of the proposed magnitude to this known measure. This proportion is always expressed by numbers; so that a number is nothing but the proportion of one magnitude to another arbitrarily assumed as the unit.
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> JG: Euler did not quite fully understand that a number is a name given to a measure that describes a magnitude or size. He focused more on a physical unit. Also, Euler did not make the connection between ratio and number.
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> 5. From this it appears that all magnitudes may be expressed by numbers; and that the foundation of all the Mathematical Sciences must be laid in a complete treatise on the science of Numbers, and in an accurate examination of the different possible methods of calculation. This fundamental part of mathematics is called Analysis, or Algebra.
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> JG: It is untrue that all magnitudes may be expressed by numbers. Moreover, Euler did not fully realise that algebra is a weaker form of geometry in which measures are performed in terms of an abstract unit (whose size is irrelevant) rather than a chosen physical unit. The properties using physical units are transferred to algebra with the introduction of incommensurable magnitudes such as pi, e, sqrt(2), etc.

One of the striking things about Euler's Elements of Algibberish is that he knows how to manipulate algebraic expressions well, which proves that not much understanding is required of today's mainstream assembly line graduates. They simply memorise the techniques and run with it.