E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like he is trying to develop some general ideas about special functions. He gives some examples of numbers he claims but does not prove cannot be represented by definite integrals of algebraic functions. Euler has the idea that if we knew more about the function with the power series $\sum x^{t_n}$ where $t_n$ is the $n$th triangular number, this could lead to a proof of Fermat's theorem that every positive integer is the sum of three triangular numbers. This doesn't end of being fruitful for Euler, but in fact later Jacobi proves a lot of results like this with his theta functions. The last paragraph (\S 9) is not clear to me. My best reading is that there are infinitely many "levels" of transcendental numbers and that this is unexpected or remarkable.
Deep Dive into On highly transcendental quantities which cannot be expressed by integral formulas.
E565 in the Enestrom index. Translated from the Latin original, “De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet” (1775). Euler does not prove any results in this paper. It seems to me like he is trying to develop some general ideas about special functions. He gives some examples of numbers he claims but does not prove cannot be represented by definite integrals of algebraic functions. Euler has the idea that if we knew more about the function with the power series $\sum x^{t_n}$ where $t_n$ is the $n$th triangular number, this could lead to a proof of Fermat’s theorem that every positive integer is the sum of three triangular numbers. This doesn’t end of being fruitful for Euler, but in fact later Jacobi proves a lot of results like this with his theta functions. The last paragraph (\S 9) is not clear to me. My best reading is that there are infinitely many “levels” of transcendental numbers and that this is unexpected or remarkable
1. Integral formulas, whose integration cannot be obtained in terms of algebraic quantities, are commonly regarded as the single source of all transcendental quantities. Thus from the integral formula dx x are born the logarithms, and from the formula dx 1 + xx circular arcs; these quantities, although transcendental, have now indeed been absorbed into Analysis, so that they can be dealt with as easily as algebraic quantities. Besides these indeed are also the quantities which involve the rectification of conic sections, which now have been so explored by Geometers that problems which lead to these can obtain perfect solutions. These transcendental quantities are further contained in the integral formulas of the type 1 dx √ f + gxx b + kxx .
Whenever other transcendental quantities occur, they always permit representation as the quadrature of certain curved lines; doing this leads to integral formulas which, though complicated, are able to express the true values of such transcendental quantities.
- I have observed nevertheless that innumerable types of transcendental quantities can be exhibited which cannot be expressed in any way by integral * Presented to the St. Petersburg Academy on October 16, 1775. Originally published as De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet, Acta academiae scientiarum Petropolitanae 4 (1784), no. II, 31-37. E565 in the Eneström index. Translated from the Latin by Jordan Bell, Department of Mathematics, University of Toronto, Toronto, Canada. Email: jordan.bell@utoronto.ca
1 Translator: This is an elliptic integral. Euler’s 1760 paper “Consideratio formularum, quarum integratio per arcus sectionum conicarum absolvi potest”, E273, looks like it’s about integrals of this form.
formulas, even if their true values can often be defined easily enough. Quantities of this type arise principally from infinite series whose sums can so far be reduced in no way to integral formulas. Especially noteworthy among these is this infinite series, differing slightly from the geometric,
the denominators of whose fractions are powers of two less unity, whose approximate sum is indeed not very difficult to assign. It is easily understood moreover that this can be expressed neither by a rational nor by irrational numbers; then indeed it also seems certain enough that it can be expressed neither by logarithms or circular arcs. So far however no path is apparent for investigating an integral formula of this type, whose value exhibits exactly the sum of this series. But letting all the terms of this series each be converted into a geometric series in the usual way, it is worth noting that its sum can be represented by the following formulas
where 1 2 xx denotes the sum of the infinite series whose term corresponding to the index x is 1 2 xx , which will therefore be
but equally as the sum of this series cannot be exhibited in any way, the same should be understood for the remaining parts. The true sum of the given series is approximately 1, 606695152; if this number were found to be equal to any known number, such as a logarithm or a circular arc, it would without doubt be an extraordinary discovery.
- Like how we have subtracted unity from the powers of two here, thus let us add these to the indefinite number x, and let us put
which equation, if x is taken for the abscissa and y for the ordinate, expresses a certain curved line in which in fact one can assign a single ordinate corresponding to the abscissa x = 0, which of course will be y = 2. But for all the other abscissas, the ordinates will be highly transcendental quantities, which thus far seem not to be able to be expressed by integral formulas, so that the nature of this curve is such that it cannot be expressed by any equation, either differential or integral. In the meanwhile however it is clear that the abscissas
lead to infinitely large ordinates, while on the other hand taking x = ∞ the ordinates will vanish.
- This equation can be made more general if we let any particular number a be assumed in place of 2, so that it would be
where the ordinate corresponding to the abscissa x = 0 is
and indeed again all the remaining terms are highly transcendental. It is also clear that if one takes a = 1, all the ordinates will then be infinite, unless perhaps the abscissa x is taken as infinite. Like how we gave all the fractions here the + sign, they can also thus alternate, so that it would be
while now the ordinate that will correspond to the abscissa x = 0 is
Moreover, in place of x its powers could be written in the following terms, so that an equation of this type would be obtained
which certainly is composed such that the nature of this curve seems not to be able to be expressed by any finite equation, either differential or integral.
- Truly besides these forms there are infinitely many others which can be exhibited, proceeding only by powers of some quantity x, which
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