Adriaan van Roomen's 45th degree equation in 1593

Question

Adriaan van Roomen proposed a 45th degree equation in 1593(see this book, picture reference as follows): $$ \begin{gathered} f(x) = x^{45} - 45x^{43} + 945x^{41} - 12300x^{39} + 111150x^{37} - \color{red}{740459}x^{35} + 3764565x^{33} \\- 14945040x^{31} + 46955700x^{29} - 117679100x^{27} + 236030652x^{25} - 378658800x^{23} \\+ 483841800x^{21} - 488494125x^{19} + 384942375x^{17} - 232676280x^{15} + 105306075x^{13} \\ - \color{red}{34512074}x^{11} + 7811375x^9 - 1138500x^7 + 95634x^5 - 3795x^3 +45x \\ = \sqrt{\frac{7}{4}-\frac{\sqrt{5}}{4}- \sqrt{\frac{15-3\sqrt{5}}{8}}} \approx 0.4158234. \end{gathered}\tag{1} $$ Another source said the right side is: $\sqrt{\frac{7}{4}-\frac{\sqrt{5}}{4}-\frac{5\sqrt{3}}{8}} $. ( EDIT: According to the answer, the coefficients in red are wrong.)

Another mathematician Viète noticed that left side is nothing but the expansion of sine using $$ \sin3\theta = 3\sin\theta - 4\sin^3\theta, $$ recursively, and the equation (1) reduced to: $$ \sin(\alpha) = c, \quad \text{ for } \alpha \in (0,\frac{\pi}{2}) \tag{2} $$ where $x = \sin(\alpha/45)$ (EDIT: According to the answer, this is not correct, should be $x = 2\sin(\alpha/45)$).

Now $2k\pi+\alpha$ satisfies (2) too, hence the solutions are: $$ x = \sin\left(\frac{\alpha + 2k\pi}{45}\right), \quad \text{ for } k = 0,1,2,\ldots,44. $$ Totally 45 roots in $(-1,1)$.

However, when drawing the graph of $f(x)-c$, something does not look quite right:

There are two other roots near roughly 3 and -3? Am I missing something here?

Answer 1

Typos! Scroll to the end to see which coefficients are typos.

$$\begin{align} \sin(45t)&=\Im\left(e^{45it}\right)\\ &=\Im\left(\left(e^{it}\right)^{45}\right)\\ &=\Im\left(\left(\cos(t)+i\sin(t)\right)^{45}\right)\\ &=\Im\left(\sum_{k=0}^{45}\binom{45}{k}\left(i\sin(t)\right)^k\cos^{45-k}(t)\right)\\ &=\frac{1}{i}\left(\sum_{k=0}^{22}\binom{45}{2k+1}\left(i\sin(t)\right)^{2k+1}\cos^{44-2k}(t)\right)\\ &=\sum_{k=0}^{22}\binom{45}{2k+1}(-1)^k\sin^{2k+1}(t)\cos^{44-2k}(t)\\ &=\sum_{k=0}^{22}\binom{45}{2k+1}(-1)^k\sin^{2k+1}(t)\left(1-\sin^2(t)\right)^{22-k}\\ &=\sum_{k=0}^{22}\binom{45}{2k+1}(-1)^k\sin^{2k+1}(t)\sum_{j=0}^{22-k}\binom{22-k}{j}(-1)^j\sin^{2j}(t)\\ &=\sum_{k=0}^{22}\sum_{j=0}^{22-k}\binom{45}{2k+1}\binom{22-k}{j}(-1)^{j+k}\sin^{2k+2j+1}(t)\\ &=\sum_{k=0}^{22}\sum_{j=k}^{22}\binom{45}{2k+1}\binom{22-k}{j-k}(-1)^{j}\sin^{2j+1}(t)\\ &=\sum_{j=0}^{22}\sum_{k=0}^{j}\binom{45}{2k+1}\binom{22-k}{j-k}(-1)^{j}\sin^{2j+1}(t)\\ &=\sum_{j=0}^{22}(-1)^{j}\left[\sum_{k=0}^{j}\binom{45}{2k+1}\binom{22-k}{j-k}\right]\sin^{2j+1}(t)\\ &=\sum_{j=0}^{22}(-1)^{j}\left[\sum_{k=0}^{j}\binom{45}{2k+1}\binom{22-k}{j-k}\right]x^{2j+1}\\ &=45x+\cdots+17592186044415x^{45} \end{align}$$

But now if we replace $x$ by $\frac{x}{2}$ and multiply the whole thing by $2$, we have

$$ \sum_{j=0}^{22}\left(\frac{-1}{4}\right)^{j}\left[\sum_{k=0}^{j}\binom{45}{2k+1}\binom{22-k}{j-k}\right]x^{2j+1} $$ which expands as $$ \begin{gathered} f(x) = x^{45} - 45x^{43} + 945x^{41} - 12300x^{39} + 111150x^{37} - 740\mathbf{2}59x^{35} + 3764565x^{33} \\- 14945040x^{31} + 46955700x^{29} - 117679100x^{27} + 236030652x^{25} - 378658800x^{23} \\+ 483841800x^{21} - 488494125x^{19} + 384942375x^{17} - 232676280x^{15} + 105306075x^{13} \\ - 3451207\mathbf{5}x^{11} + 7811375x^9 - 1138500x^7 + 95634x^5 - 3795x^3 +45x \\ \end{gathered} $$ and the coefficients of $x^{35}$ and of $x^{11}$ are typos.

Note that $x=2\sin(\alpha/45)$ is the substitution, not $x=\sin(\alpha/45)$, and that we multiplied by $2$, so that doubles the value of $c$.

Answer 2

The detective work for this, mathematical and textual, may be worth recording.

The mathematical reasoning in the question is correct, so the only possibility of not having the stated roots is if the polynomial were wrong. And if it were incorrect, this would probably involve some error in the middle terms, which for $|x|<1$ would be indetectible from a graphical plot, but where the error can become significant outside $(-1,1)$. The suspicion of a typo is raised, as was suggested in the comments below the question and confirmed by a recalculation of the polynomial in alex.jordan's answer.

The polynomial in the question is not the one with $\sin (45 \alpha)$ computed from $x=\sin (\alpha)$. That one is not monic and has a leading coefficient of $2^{44}$. Online preview does not display the equation (page 46), but a search for "van Roomen" within the book shows (notes, p. 186) that $x$ in the polynomial is $2 \sin \alpha$ and from that and the discussion there one can see that the displayed polynomial computes $2 \sin 45 \alpha$ from $x$. This polynomial has an expression

$P_{45}(x) = 2 \sin (45 \arcsin \frac{x}{2})$

that can be expanded by software

http://www.wolframalpha.com/input/?i=2+sin(45+arcsin+(x/2))

http://www.wolframalpha.com/input/?i=FunctionExpand(2+Sin(45+ArcSin(x/2)))

$P_{45}(x) $ = x^45-45 x^43+945 x^41-12300 x^39+111150 x^37-740259 x^35+3764565 x^33-14945040 x^31+46955700 x^29-117679100 x^27+236030652 x^25-378658800 x^23+483841800 x^21-488494125 x^19+384942375 x^17-232676280 x^15+105306075 x^13-34512075 x^11+7811375 x^9-1138500 x^7+95634 x^5-3795 x^3+45 x .

The polynomial in the question differs from this by $-200 x^{35} + x^{11}$. The discrepancy is $-199$ at $x=1$ and $+199$ at $x=-1$, as seen in the first graph. The second graph is a form of Runge's phenomenon; the wrong polynomial can be seen as a high degree interpolating polynomial of a function (the discrepancy) that is close to zero on most of the interval $(-1,1)$ and is sampled at points where it is small, but the interpolant has sharp oscillations at the endpoints.

The polynomial is correct in the original references

van Roomen (1593, http://books.google.com/books?id=YyA8AAAAcAAJ , end of Prefatorio)

Viete (1595 , http://books.google.com/books?id=XxQ8AAAAcAAJ )

and in the modernized discussion in

Tignol (2001, http://books.google.com/books?id=hO6HYckIYxsC&pg=PA30 )

The bug is in symmetrically located coefficients $x^{35}$ and $x^{11}$ which suggests the possibility of a systematic error, such as incorrect copying of a binomial coefficient used in a formula. No such error is easily detected. The expression of $P_{45}(x)$ as a Chebyshev or Dickson polynomial (given in Tignol's book, or at Wikipedia as $D_{45}(\cdot, -1)$ ) uses binomial coefficients multiplied by rational factors, $\frac{n}{n-p} {{n - p} \choose p} x^{n-2p}$ that at the necessary values of $n$ and $p$ would create non-integers for any error in the binomial coefficient and the same from any small change to the factor. The summation in alex.jordan's answer does not look like any small modification could cause the error of $200x^{35}$.

Pesic's book (page 186) cites an older article

Guido Vetter, "Sur l'equation du quarante-cinquieme degre d'Adriaan van Roomen," Bulletin des sciences mathematiques (2) 54, 277-283 (1954)

which would be the first place to look to see if the error is new or copied.

Eli Maor's popular book Trigonometric Delights avoids the whole problem by listing only the first and last few coefficients in the polynomial and a " ... " in between, in line with the first principle of popular science editing: every scary-looking mathematical formula will cut the sales in half.