The "secret formula" is a method found by Tartaglia to solve cubic equations. The 16th century priority fight between Tartaglia and Cardano's student Ferrari over this formula is well known. A brief summary: Niccolo Tartaglia, an abbaco master in Venice, was challenged in a mathematical duel by Antonio Fior, and solved the questions, all reducible to solving cubic equations, in a couple of hours. This made him instantly famous, since Luca Pacioli had stated that, contrary to the quadratic equation, there did not exist a general formula for cubic equations. Tartaglia did not want to reveal his secret formula, probably to keep up his fame and to have an advantage in future duels. Gerolamo Cardano however wanted to include the formula in his book on algebra that he was preparing. Only after a lot of nagging, Tartaglia discloses to him his method in verse form, subject to strict secrecy. When Cardano and his student Ludovico Ferrari discovered that the formula was known in unpublished work from Scipione dal Ferro, they considered the promise to Tartaglia not binding any more, and the method was included in the book with proper reference. This resulted in a public fight between Tartaglia and Ferrari publishing insulting pamphlets back and forth.
Niccolo Tartaglia was born in Brescia around 1500. His last name is often believed to be Fontana, but Toscano claims that there is no solid proof for that. Niccolo preferred to use his nickname Tartaglia, i.e. the stammerer, because that is what he did since at the age of 12, a French soldier's sabre mutilated his jaw and left him for dead during a reprisal attack of the troupes of Louis XII on his home town. Against all odds, his mother could keep him alive. Later he became an abbaco teacher. That means that he instructed and applied the practical use of numerical calculation using the newly introduced Hindu-Arabic numerals as described in Fibonacci's Liber Abbaci, rather than the impractical Roman numerals. This practical kind of mathematics was needed in commerce for bookkeeping or for example to converse different measures or currency. It was quite different from the geometry of Euclid's Elements that was taught at an academic level.
The Renaissance habit of having public challenges and scientific duels had some historical background. A number of questions were formulated by the challenger to be solved within a certain time span. It was a gentlemen's agreement though that no questions should be asked that the challenger was not able to solve himself. The one that was challenged then reposted with a set of questions for the challenger, and the winner of the duel was the one who first solved all the problems first or who solved most of the problems. Tartaglia received in 1530 two questions by Zuanne de Tonini da Coi, and that surprised him because the problems reduced to the solution of a cubic equation, which was claimed to be impossible by Pacioli. So he assumed that da Coi did not know how to solve it either. Nevertheless he started thinking about the problem, and obviously found a solution, at least for some cases. It should be noted that what we write in modern notation as $x^3+ax=c$ and $x^3=bx+c$ were considered to be two different types of equations. Because the terms represented (positive) quantities, (often lengths with a geometric interpretation), they could not be zero or negative. Only positive coefficients were allowed, which made it difficult to switch terms to the other side of the equal sign.
To explain the state of the art of algebraic manipulation, Toscano sketches in a second chapter the history of how algebra came to Europe from the Babylonian Plimpton 322 tablet and the Egyptian Rhind papyrus to Al-Khwarizmi's Algebra book (The Compendious Book on Calculation by Completion and Balancing) in which is explained how to solve equations without explicitly switching terms to the other side of the equal sign. It would have been simpler if negative numbers were allowed and if our symbolic notation was used. Although the latter was once promoted by Diophantus of Alexandria (3rd century), the habit was lost over time. It is only because of the 16th century events described here that our modern notation and manipulation came about.
The next chapter is describing the 1535 duel with Fior that started Tartaglia's fame. Antonio Fior challenged Tartaglia with problems that all reduced to cubic equations and Tartaglia, who had figured out how to do it since da Coi's questions, gave the answers in a few hours long before Fior could solve one of Tartaglia's problems. Whether Fior was able to solve the problems himself, is not clear since he kept begging Tartaglia to reveal his method, although he claimed that the method was explained to him by "some mathematician" 30 years ago. This was most likely dal Ferro since Fior was his assistant.
But Tartaglia was now also approached by Cardano, first through his publisher, who wanted to include Tartaglia's method in his book on algebra. When Cardano invited him later to Milan, Tartaglia finally disclosed his method after Cardano had sworn not to publish it. Toscano explains the rhyme that Tartaglia used to summarize how to solve the two forms of the cubic equation mentioned above and how a third form is reduced to one of those. When Cardano's book Ars Magna sive de regulis algebraicis was published in 1545 it was a big success and historians consider it as the beginning of modern mathematics. It contained, besides Ferrari's solution for the fourth degree equation, also the method for the cubic equation with proper reference to Dal Ferro and Tartaglia. Tartaglia was however furious and he published his Quesiti et inventioni nuove containing his account of what has happened, alongside some insulting remarks about Cardano. This was published one year after Cardano's book, while he had neglected for many years to publish his method himself. Cardano had accepted a long-cherished physician's position and had left mathematics teaching, so Ferrari took up the defence of his supervisor and there was quite some verbal abuse in public pamphlets exchanged between Ferrari and Tartaglia in subsequent months. This culminated in a duel between both in 1548 in Milan, with Ferrari as victor.
Cardano's book is important for the history of mathematics because it initiated some ideas that lead to complex numbers. On the other hand, the Cardano-Tartaglia-Fior-Ferrari interaction is a juicy topic that easily lends itself to be discussed in popular science books. So it has been told by many authors, but it is often Cardano who is placed at the center of the story. For example in P.J. Nahin An Imaginary Tale: The Story of √-1 (1998/2010) some time is devoted to this melee and in the novel by M. Brooks The Quantum Astrologer's Handbook (2017) Cardano is the main historical character. In the present book, it is basically the same story all over, but told more from Tartaglia's viewpoint. A lot is taken from Tartaglia's own account with many translated quotations in which the mutual scolding in the pamphlets are made blatantly clear. There is of course some background and history of mathematics but Toscano's main focus is the solution of the cubic equation leaving some other work of Tartaglia and Cardano in the shadow. For example Tartaglia wrote a treatise on ballistics and found that the maximum reach was obtained firing in an angle of 45 degrees. The result is correct although it has several mistakes for which Ferrari reproached him later. He also translated Euclid's Elements to Italian (working on this was his excuse for not publishing his formula).
Toscano has a pleasant writing style (and/or the translation by Arturo Sangalli is smooth). The opener of the books describes Niccolo with his mother and sister lost in the chaos of French soldiers attacking Brescia. Niccolo is hit twice by the sabre of a soldier and left for dead. That is like the opener of a dramatic novel. The attention of the reader is immediately caught. As Toscano unravels the historical development, he makes use of many quotations, which are fortunately provided by Tartaglia himself. This implies that the story is told close to how Tartaglia has experienced what has happened. However Toscano does not hesitate to give some interpretations and place some question marks where appropriate. The yeast of the story has been told already many times, but it has never been told like Toscano does in this book.