This is an EDIT version of my original question:
Recently I've been interested in the history of the Theory of Equations. The thing is that I learned about this mathematician named John Colson, he published a very interesting paper: Aequationum Cubicarum & Biquadraticarum, tum Analytica, tum Geometrica & Mechanica, Resolutio Universalis in the Philosophical Transactions, he was a contemporary of De Moivre, althought he published his infamous formula in 1722, also in the Philosophical Transactions (I made a question about it before) and Colson in 1706.
Now, on to the paper, remember, I'm not fluent in latin, so there might be things that I miss.
He presents his work with the three roots of a given universal cubic equation, & the way he presents them is by representing them as linear combination of the roots of unity, first he shows with 7 examples that considering the roots as such, works!
Subsequently to the examples, he actually shows how he got that the roots can be written like that. I was thinking of writing down all the procedure, but is very long, and is actually very understandable from the paper. However, if you want to know my interpretation from a particular sentence, I'm pretty much done with the translation (just the cubic part), just remember I'm not fluent in latin, & probably I committed big mistakes with the interpretation.
I consider his work to be very original (diferent) for his time, I don't know when the use of roots of the unity was introduced, probably before him, but he definitely gave them some play, since (almost) everything related to solving those equations seemed very geometrically based, i.e., a lot of the things people would use to solve the cubic, for example, were derived from geometric properties, we can see that he works the other way around, remember the name of the article: "Universal solution of the biquadratic and cubic equations, both analytical and geometrical and mechanical", from algebra he gets the geometry.
I don't know if I'm making much sense, but let see if this picture helps. I belive this is something Descartes said, I don't remember very well, but this was the way people of Colson's time used to think.
What this represents, is that if you have a problem in geometry, then you can represent it with algebra, and if you have a problem in algebra, then it belongs to a problem in geometry. I hope this helps to illustrate my point.
(BTW, Galois, Abel and others later showed us that this not true)
Now the point of my question is, if Colson could represent the start of the independence of algebra from geometry. If we look at the big picture, and since many of us were born in the XX century, we know how this is going to end, so would it be so naive of my part to consider Colson as this kind of hero? Who could be a better representative for this?
Thanks!
Best Answer
Chapter 5 (Roots as sums of radicals, pp. 104-120) of Stedall's book (bibliographic information below) discusses:
Jacqueline Anne Stedall, From Cardano's Great Art to Lagrange's Reflections: Filling a Gap in the History of Algebra, Heritage of European Mathematics, European Mathematical Society, 2011, xii + 224 pages. MR 2012a:01012; Zbl 1231.01006
Excerpt from pp. 104-106 of Stedall's book
(ADDED NEXT DAY) Below are some books that you might also want to know about. I do not think Colson is mentioned in any of these books, but given what you've written in your question and in your follow-up comments, I would encourage you to consider getting a copy of some or all of these books.
[1] Isabella G. Bashmakova and Galina S. Smirnova, The Beginnings and Evolution of Algebra, translated from the Russian by Abe Shenitzer, The Dolciani Mathematical Expositions #23, Mathematical Association of America, 2000, xvi + 179 pages. MR 2000h:01002; Zbl 942.01001 [See also here.]
[2] Roger Lee Cooke, Classical Algebra. Its Nature, Origins, and Uses, John Wiley and Sons (Wiley-Interscience), 2008, xii + 206 pages. MR 2009b:00001; Zbl 1139.00001 [Review at MAA website.]
[3] Jacques Sesiano, An Introduction to the History of Algebra. Solving Equations from Mesopotamian Times to the Renaissance, Mathematical World #27, American Mathematical Society, 2009, viii + 174 pages. MR2514537; Zbl 1182.01002 [See also here.]
[4] Jacqueline Anne Stedall, A Discourse Concerning Algebra: English Algebra to 1685, Oxford University Press, 2002, xii + 294 pages. MR 2005c:01015; Zbl 1035.01006 [Review at MAA website.]
[5] Veeravalli Seshadri Varadarajan, Algebra in Ancient and Modern Times, Mathematical World #12, American Mathematical Society, 1998, xvi + 142 pages. MR 99d:01007; Zbl 917.01002 [See also here.]
[6] Girolamo Cardano, The Great Art or the Rules of Algebra, translated and edited by T. Richard Witmer, MIT Press, 1968, xxiv + 267 pages. MR 40 #4074 and 50 #12562; Zbl 191.27704 [Reprinted by Dover Publications in 1993 (MR 94k:01038; Zbl 862.01034) and in 2007.]
[7] François Viète, The Analytic Art, translated by T. Richard Witmer, The Kent State University Press, 1983, i + 450 pages. MR 86b:01012; Zbl 558.01041 [Reprinted by Dover Publications in 2006 (Zbl 1115.01017).]
[8] Edward Waring, Meditationes Algebraicae. An English Translation of the Work of Edward Waring, edited and translated from the Latin by Dennis Weeks, American Mathematical Society, 1991, lx + 459 pages. MR 93a:01026