"(1) In analysis the word 'equation', standing by itself, means an equality properly constructed in accordance with [the rules of] zetetics. (2) Thus an equation is a comparison of an unknown magnitude and a known magnitude." (François Viète, "On the Meaning and Components of Analysis and on Matters Useful to Zetetics", 1591)
"It is true that not every geometric construction is elegant, for each particular problem has its own refinements. It is also true that [that construction] is preferred to any other that makes clear not the structure of a work from an equation but the equation from the structure; thus the structure demonstrates itself. So a skillful geometer, although thoroughly versed in analysis, conceals the fact and, while thinking about the accomplishment of his work, sheds light on and explains his problem Then, as an aid to the arithmeticians, he sets out and demonstrates his theorem with the equation or proportion he sees in it." (François Viète, "On the Meaning and Components of Analysis and on Matters Useful to Zetetics", 1591)
"Reduction is a common division45of the homogeneous magnitudes making up an equation by the given magnitude by which the highest grade of the unknown is multiplied so that this grade may lay claim to the title of power by itself and that from this an equation [in proper form] may finally remain." (François Viète, "On the Meaning and Components of Analysis and on Matters Useful to Zetetics", 1591)
"The [analytic] art teaches, in addition, the resolution of [all] powers whatsoever, whether pure or affected, [this last being] something understood by neither the old nor the new mathematicians." (François Viète, "On the Meaning and Components of Analysis and on Matters Useful to Zetetics", 1591)
"There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered. Theon called it analysis, which he defined as assuming that which is sought as if it were admitted [and working] through the consequences [of that assumption] to what is admittedly true, as opposed to synthesis, which is assuming what is [already] admitted [and working] through the consequences [of that assumption] to arrive at and to understand that which is sought." (François Viète, "On the Meaning and Components of Analysis and on Matters Useful to Zetetics", 1591)
In geometry,6 to be sure, the accident of a fraction or an irrational does not usually prevent equations from being solved readily. nor does the imperfection of a negative, for the subject on which the geometer works is always certain. But multiplicity of affections8 is a hindrance, and the higher the power and the order of an affection, the more likely it is that a fraction or surd9 will appear in the solution of a problem." (François Viète, "On the Structure of Equations as Shown by Zetetics, Plasmatic Modification and Syncrisis", 1646)
"Will an analyst attempt [the solution of] any proposed equation without understanding how it is composed so that he can avoid the rocks and crags? Will he [be able to] transpose, depress, raise and generally work with sureness like an expert anatomist if, by some new discovery of zetetics, an unknown is proposed in terms other than those [originally] given but with a given difference from or ratio to that which was proposed?Above all, the origin of equations and their fundamental structure is worthy of being understood by the analyst who strives for and pursues that expertise by which the way of reduction opens itself to him." (François Viète, "De Aequationum Recognitione et Emendatione Tractatus Duo", 1646)
"Syncrisis is the comparison of two correlative equations in order to discover their structure. Two equations are said to be correlatives when they are similar and, in addition, have the same given magnitudes both for the coefficients of their affections and for their homogeneous terms of comparison.66 Their roots, nevertheless, are different either because their structure is such that they may be solved by two or more roots or because the quality or sign of their affections is different." (François Viète, "De Aequationum Recognitione et Emendatione Tractatus Duo", 1646)
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