François Viète


Source: The Analytic Art, trans. by T. Richard Witmer, Kent State U. Press, 1983.
 

Introduction to the Analytic Art


Chapter I
On the Meaning and Components of Analysis and on Matters Useful to Zetetics

    There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered.  Theon 1  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.
    Although the ancients propounded only two kinds of analysis, zetetics and poristics, to which the definition of Theon best applies, I have added a third, which may be called rhetics or exegetics.  It is properly zetetics by which one sets up an equation or proportion between a term that is to be found and the given terms, poristics by which the truth of a stated theorem is tested by means of an equation or proportion, and exegetics by which the value of the unknown term in a given equation or proportion is determined.  Therefore the whole analytic art, assuming this three-fold function for itself, may be called the science of correct discovery in mathematics.2
    Now whatever pertains to zetetics begins, in accordance with the art of logic, with syllogisms and enthymemes 3  the premises of which are those fundamental rules with which equations and proportions are established.  These are derived from axioms and from theorems created by analysis itself.  Zetetics, however, has its own method of proceeding.  It no longer limits its reasoning to numbers, a shortcoming of the old analysts, but works with a newly discovered symbolic logistic which is far more fruitful and powerful than numerical logistic for comparing magnitudes with one another.  It rests on the law of homogeneous terms 4  first and then sets up, as it were, a formal series or scale of terms ascending or descending proportionally from class to class 5  in keeping with their nature and, [by this series,] designates and distinguishes the grades and natures of terms used in comparisons.

Chapter V
On the Rules of Zetetics

    The manner of working in zetetics is, in general, contained in these rules:
 

  1. If it is a length that is to be found and there is an equation or proportion latent in the terms proposed, let x be that length.
  2. If it is a plane that is to be found and there is an equation or proportion latent in the terms proposed, let x2  be that plane.
  3. If it is a solid that is to be found and there is an equation or proportion latent in the terms proposed, let x3  be that solid.
What is to be found will, in short, rise or fall, in keeping with its nature, through the various grades of the magnitudes of comparison.
  1. Magnitudes, both given and sought, are to be combined and compared, in accordance with the given statement of a problem, by adding, subtracting, multiplying and dividing, always observing the law of homogeneous terms.
Hence it is evident that in the end something will be found that is equal to the unknown or one of its powers.  This may be made up entirely of given terms or it may be the product of given terms and the unknown or of those terms and a lower-order grade.
  1. In order to assist this work by another device, given terms are distinguished by unknown by constant, general and easily recognized symbols, as (say) by desginating unknown magnitudes by the letter A and the other vowels E, I, O, U and Y and given terms by the letters B, G, D and the other consonants.
  2. Terms made up exclusively of given magnitudes are added to or subtracted from one another in accordance with the sign of their affection and consolidated into one. 6   Let this be the homogeneous term of comparison or the constant and put it on one side of the equation.
  3. Likewise, terms made up of different quantities and the same lower-order grade are added to or subtracted from one another in accordance with the sign of their affection and consolidated into one.  Let this be the homogeneous term of affection or the lower-order homogeneous term.
  4. Keep these lower-order homogeneous terms with the power they affect or by which they are affected and place them and the power on the other side of the equation.  Hence the constant term will be designated in keeping with the nature and the order of the power.  It will be called pure if [the power] is free from affection.  But if [the power] is accompanied by homogeneous terms of affection, show this by the [proper] symbols of affection and of degree along with any supplementary terms that are their coefficients.
  5. If the constant happens to be associated with a subordinate homogeneous term, carry out a transposition.  Transposition is a removal of affecting or affected terms from one side of an equation to the other with the contrary sign of affection.  That an equation is not altered by this operation is now to be demonstrated:

    6.  

     
     





    Proposition I
    An equation is not changed by transposition.

Let  A2-Dp = G2 - BA.7   I say that A2 + BA = G2 + Dp  and that the equation is not changed by this transposition with contrary signs of affection.  For since  A2-Dp = G2 - BA, add  Dp + BA  to both sides.  Then by common agreement  A2-Dp + Dp + BA= G2- BA + Dp + BA.  The negative affection on each side of this equation cancels a positive: on one side the affection Dp vanishes, on the other the affection BA.  This leaves A2 + BA = G2 + Dp.
 
  1. If it happens that all the magnitudes given are multiplied by a grade and that, therefore, no pure constant term is immediately apparent, carry out a depression.  Depression is an equal lowering of the power and the lower-order terms in the observed order of the scale until the lowest variable term becomes a pure constant to which the others can be compared.  That an equation is not changed by this operation is now to be demonstrated:
Proposition II
An equation is not changed by depression.

Let A3 + BA2 = ZpA.  I say that by depression A2 + BA = Zp, for all of these solids have been divided by a common divisor, [a process] that, it has been settled, does not change an equation.

  1. If it happens that the highest grade of the unknown does not stand by itself but is multiplied by some given magnitude, carry out a reduction.  Reduction is a common division of 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.  That an equation is not impaired by this operation is now to be demonstrated:
Proposition III
An equation is not changed by reduction.

Let BA2 + DpA = Zs. 8   I say that by reduction

for all the solids have been divided by a common divisor, [a process] which, it has been settled, does not change an equation.

  1. Following all this, an equation may be said to be clearly expressed and in proper order.  [It may be] restated, if you wish, as a proportion, but with this particular warning: the product of the extreme terms [of the proportion] corresponds to the power plus the homogeneous terms of affection and the product of the mean terms corresponds to the constant.
  2. Hence a properly constructed proportion may be defined as a series of three or four magnitudes so expressed in terms, either pure or affected, that all of them are givens except the one that is being sought or its power and its lower-order grades.
  3. Finally, when an equation or proportion has been set up, zetetics may be said to have fulfilled its task.
Diophantus used zetetics most subtly of all in those books that have been collected in the Arithmetic.  There he assuredly exhibits this method in numbers but not in symbols, for which it is nevertheless used. 9   Because of this his ingenuity and quickness of mind are the more to be admired, for things that appear to be very subtle and abstruse in numerical logistic are quite familiar and even easy in symbolic logistic.
 
 

Preliminary Notes in Symbolic Logistic



[...]

Proposition VI
To add the difference between two magnitudes to their sum.

Let A + B be added to A - B.  The sum is 2A.  Wherefore,

Theorem
The sum of two magnitudes plus their difference is equal to twice the greater magnitude.

Proposition VII
To subtract the difference between two magnitudes from their sum.

Let A - B be subtracted from A + B.  The remainder is 2B.  Wherefore,

Theorem
The sum of two magnitudes minus their difference is equal to twice the smaller magnitude.

Propostion VIII
If the same magnitude is diminished by unequal subtrahends, to subtract one [quantity] from the other.

Let A - E be subtracted from A - B.  The remainder will be E - B.  Note that this is also the difference between the subtrahends.  Wherefore,

Theorem

If a magnitude be diminished by unequal subtrahends, the difference between the remainders is the same as the difference between the subtrahends.

Proposition IX
If the same magnitude is increased by unequal addends, to subtract one [quantity] from the other.

Let A + Bbe subtracted from A + G.  The remainder will be G - B.  Wherefore,

Theorem

If the same magnitude is increased by unequal addends, the difference between the sums is the same as the difference between the addends.

Proposition X

If the same magnitude is increased by an addend and decreased by a subtrahend, which are unequal, to subtract one [quantity] from the other.

Let A - Bbe subtracted from A + G.  The remainder will be G + B.  Whence

Theorem

If the same magnitude is increased by an addend and decreased by a subtrahend which are unequal, the difference between the sum and the remainder is equal to the sum of the addend and the subtrahend.

Proposition XI
To construct a pure power from a binomial root.

Let there be a binomial root, A + B.  A pure power is to be constructed from it.
    First, let the square be constructed.  Since a root multiplied by itself makes a square, mulitply A + B by A + Band collect the individual planes that result.  These will be A2 + 2AB + B2 which are, accordingly, equal to the square of A + B.
    Secondly, let the cube be constructed.  Since a root multiplied by its square makes a cube, let A + B be multiplied by the square of A + B just set out and collect the individual solids that result.  These will be A3 + 3AB2 + 3A2B+ B3 which will, therefore, be equal to the cube of A + B.
    Third, let the fourth power be constructed.  Since a root multiplied by its cube makes the fourth power, multiply A + B by the cube of A + B just shown and collect the individual plano-planes that result.  These will be A4 + 4AB3 + 2A2B2 + 4A3B + B4 which will, accordingly, be equal to the fourth power of A + B.
    Fourth, let the fifth power be formed.  Since a root multiplied by its fourth power makes a fifth power, multiply A + B by the fourth power of A + B just shown and collect the individual plano-solids that result.  These will be A5 + 5AB4 + 10A2B3 + 10A3B2 + 5A4B + B5 which will clearly be equal to the fifth power of A + B.
    Fifth, let the sixth power be constructed.  Since a root multiplied by its fifth power makes a sixth power, multiply A + B by the the fifth power of A + B just shown and collect the individual solido-solids that result.  These will be A6 + 6AB5 + 15A2B4 + 20A3B3 + 15A4B2 + 6A5B + B6 which will be equal, therefore, to the sixth power of A + B.
    The construction of any higher power will be no different.  From these [examples], therefore, theorems that are worth while for the whole of logistic and useful in zetetics can be derived and comprehended by a uniform method.10
 
 


 
 


Theorem I

The product of the difference between two roots and the individual homogeneous terms, taken once each, of which a power of the sum of the roots consists is equal to the difference between the next higher powers [of those roots]. 11   Hence

Corollary

The difference between these powers divided by the difference between the roots is one each of the individual homogeneous terms of which the next lower power of the sum of these roots consists.  And, contrariwise, the difference between these powers divided by the individual homogeneous terms, taken once each, of which the next lower power of the sum of these roots consists is the difference between the roots. 12

Theorem II

The product of the sum of two roots and the individual homogeneous terms, taken once each, of which a power of the difference between the roots consists is equal to the sum or the difference between the next higher powers [of those roots]--the sum if the number of individual homogeneous terms is uneven, the difference if the number of individual homogeneous terms is even. 13   Hence

Corollary

The sum or difference between [two] powers divided by the sum of their roots is one each of the individual homogeneous terms of which the next lower power of the difference of those roots consists. 14

Another Corollary

If the the number of individual homogeneous terms is uneven, of which a power of the sum of or difference between the roots consists is even, as the difference between the roots is to their sum, so the individual homogeneous terms, taken once each, of which the power of the difference between these roots consists will be to the individual homogeneous terms. taken once each, of which the same power of the sum of these roots consists. 15
 
 

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