Some perceivable inconsistencies, about whether the embedded worksheets can be viewed.

One of the facts which I did mention in an earlier posting, concern worksheets which I sometimes embedded into my postings as ‘<iframe>s’. In reality, there could be two reasons, why such worksheets fail to display in any one person’s browser:

  1. I could have embedded them, specifying (insecure) URLs that begin with ‘http://’, even though the reader may be visiting my blog, using a (secure) URL that begins with ‘httpS://’. Or,
  2. The posting might suggest that the reader “may need to” enable JavaScript from the domain ‘’, but, in certain cases this is not needed, while in other cases, it is.

‘Problem 1′ above is harder for the reader to fix at their end. It can be solved by fetching my posting using the insecure ‘http://’ URL, or, as I’ve done in a very recent posting, I could decide to embed the ‘<iframe>’ using a URL, which does not specify my domain-name, and just assumes that it’s to be the same, as the domain-name of my site.

‘Problem 2′ above represents a contradiction which may confuse some readers. Sometimes, when I generated an HTML version of a worksheet, I did this by simply clicking on a button, in the GUI of my software. In such cases, the HTML will require that the JavaScript be enabled. But, in certain other cases, all I really did was, to output the worksheet as a LaTeX document, and then to run a custom script on it, to generate other types of documents, one of which would have been an HTML document.

If I ran my custom script on some generic LaTeX document, that generates ‘HTML with equations’, it will do so in the form of ‘a hybrid document’. The resulting document is hybrid, because it can then be viewed in one of two ways:

  1. Using the built-in support for ‘MathML’, that some readers’ browsers have. This is less likely, because I only know of one browser that actually support it: ‘Firefox’. If the reader is using ‘MS Edge’ or the ‘Chrome’ browser to view my posting, I know that they do not support MathML as a built-in feature. But,
  2. Such a hybrid HTML document can also be viewed, by enabling the JavaScript that gets referred to as ‘MathJax’, and which essentially allows other browsers, including ‘MS Edge’ and ‘Chrome’, to view the equations.

I try to accommodate as many possible configurations of the readers’ browsers as I can, but the unfortunate reality is, that merely pressing the GUI button within my application, may generate HTML which is highlighted more nicely, but which is not a hybrid HTML document.



A better way to use “Maxima” to solve cubic equations…

One of the facts which I’ve pointed out in earlier postings was, that if I give the free Computer Algebra System (‘CAS’) named ‘Maxima’, a cubic equation to solve, as in, to find the exact analytical solutions to, it will fail to produce intelligible output, if the usage was naive.

More specifically, cubic equations exist that have 3 distinct, real, irrational roots, and which a ‘CAS’ should be able to solve, just because their general solution is publicly known. That solution boils down, to deriving a second cubic, which is called a ‘depressed cubic equation’, and then performing a trigonometric substitution. (:1)

A fact which I’ve also known for some time is that, especially if a person is using a free or open-source CAS, then in some cases its behaviour has not been made particularly user-friendly, in that work needs to be done by the user, to set up his or her problem for the ‘CAS’ to solve. This latter observation casts a shadow of doubt, over the question of whether a ‘CAS’ will ultimately lead experienced Mathematicians to new discoveries in Algebra, or whether this can only reduce the workload in certain situations.

In this posting I’m going to show, how ‘Maxima’ can be coerced into giving correct answers, by users who know how. What I tend to use is a Graphical Front-End to ‘Maxima’, that is itself named ‘wxMaxima’, but which has equal capabilities, except for the abilities to typeset its solutions, as well as to export its Worksheets to PDF as well as HTML format, using LaTeX.

The following embedded worksheet will only display properly in the reader’s browser, if

  • The reader has allowed JavaScript from my blog to run on his browser, and
  • The reader has also allowed JavaScript to run from a CDN named ‘’.



What’s observable here is the fact that the package ‘odes’ can be loaded, which is mainly used to solve Ordinary Differential Equations, and that afterwards, the function ‘solvet()’ can be used, even to solve certain polynomials – better than what Maxima can solve on its own, with the built-in ‘solve()’ function. (:3)

(Updated 6/14/2020, 0h30… )

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