John D. Blanton's Translation of Euler's Introductio

Errata, Comments, Questions.

P. v. Euler's Preface begins with a wonderful quotation:

Often I have considered the fact that the most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art.

Who were these "students" that Euler refers to? He gives no indication, so as we read the book, we should think about who the intended audience was.

A little later on the same page he refers to "the ordinary treatise on the elements of algebra." What are these?

P. xi.  In the last paragraph of the Preface, after noting that he will not give credit to every place to those "who have toiled herein" Euler remarks that "If the history of each problem had been discussed, that would have increased the size of this work beyond reasonable bounds." Finally he notes that some things are "entirely new."

This sets a project for we historians: Where should we add footnotes to the earlier work? What things are entirely new? And, to add yet another question, what work did the Introductio inspire?

P. vii. In the Translator's Introduction, Blanton comments about the plural "infinitorum" in the title. He understands it "to refer to the three main topics: infinite series, infinite products, and continued fractions." Do you agree?  Would you refer to an infinite series as an "infinity"?

Blanton does not indicate which edition of the Introductio he used in his translation. Because errors noted in the Opera omnia (1922l) are not corrected here, it seems that he used the Latin original or, more likely, the Belgian photographic reprint (1967l). Evidence is provided by §§ 19, 107, 126.

P. 4, l. 10, §5. "there" should be "these".

P. 12, l. 10^- and 9^-,  §19. The equation should be "Z² = az⁴Z + bz²". Blanton just repeats what is in the original; the Opera omnia (1922l) silently corrects the error (which today is considered poor editorial policy. In line 9^- Blanton incorrectly corrects Euler. Blanton has "two-valued function" but it should be "two-valued even function".

P. 26, l. 11, §39. Equation II should be -2a + b + 2c + d = -2. The original is correct.

P. 29, l. 9, §41. "A/Z" should be "A/z".

P. 67, l. 5^-, §83. "one degree" should be "degree one". 

P. 77, l. 10, §99. "cannnot" should be "can not".

P. 78, l. 5 and 6,  §101. On line 5, "z" should be "y" and on line 6, "y" should be "x".

P. 78, l. 1^-, §102. Euler denotes the logarithm of y by "l y" not "log y" as Blanton does.

P. 79, l. 3^-, §104. "number" should be "numbers".

P. 79, l. 1^-, §104. "log vy = x + y" should be "log vy = x + z".

P. 80, l. 2, §104. "logarithms" should be "logarithm".

P. 82, l. 7^-, §106. "Vlasc" should be "Vlacq".

P. 83, l. 2, §107. "p/q is constant" should be "q/p is constant".

P. 88, l. 5, §111. In Example II "usurious rate of five percent" is a poor translation. "at the usury or interest rate of five percent" would be better. At the time the word "usury" was used for "rate of interest."

P. 88, l. 7^-, §111. "n^x" should be "n^xa".

P. 90, l. 6^-, §113. "than the number" should be "then the number".

P. 95, l. 1, §118. "an finite" should be "a finite". 

P. 99, l. 9, §124. "logarithms" should be "logarithm".

P.102, l. 8^-, §126. The 113th digit of π should be "8" not "7". So it should be "1328230". The error is in the original, but is corrected in the Opera omnia (1922l).

P.102, l. 8^-, §126. "we will use the symbol π" should be "I will use the symbol π", indicating that Euler believes he is introducing the symbol π for the semi-circumfrence of a unit circle. The Latin word is "scribam" which is first person.

P. 146, l. 4, §175. Blanton fails to give one of Euler's series, namely

    5π⁵/1536 = 1 - 1/3⁵ + 1/5⁵ - 1/7⁵  + 1/9⁵ - &c.

As usual, Euler ends the series with "&c", i.e., et cetera, rather than with the ellipsis that Blanton uses. Also, in line 6, "equal" should be "even".

P. 261, l. 2^-, §306. The product should be followed by an ellipsis.

P. 284, l. 10^-, §334. Euler has a notational problem here. The original has A, B, C, D in the numerators of the partial fractions. In the Opera omnia this is silently corrected to fractur letters. Here the problem is only half solved. On the following page, in lines 4 and 5 he uses Euler's "A" instead of his "U" (once in line 4, twice in line 5).

P. 284, l. 8^-, §334. "Pxⁿ" should be "PZⁿ"