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A Source Book in Mathematics, 1200-1800

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Author: René Descartes | Date: 1897–1918

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René DescartesDescartes carried out his ideas on the algebraic representation of geometrical quantities in his Géométrie, published as appendix I to his Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences (Discourse of the method for conducting reasonwell and seeking the truth in the sciences; Leiden, 1637); Oeuvres de Descartes, ed. C. Adam and P. Tannery (12 vols.; Paris, 1897–1918), VI, 1–78, 367–485. In this Géométrie he showed how the "Algebra of the Moderns," that is, the Renaissance algebra of Cardan and his successors, can be applied to the "Geometry of the Ancients," that is, to the Greek geometry of Apollonius and Pappus. How he did it we show in Selection III.4. But at the same time, in Book Three of the Géométrie, he applied it to the theory of equations, adding new discoveries of his own in a presentation of the subject that begins to have a modern look, since his authority made it acceptable to a growing number of mathematicians. He presents some important theorems that are valid for the case in which the right-hand member of an equation is zero, such as the proposition that is a factor of the left-hand member if is a root of the equation. His text also contains a formulation of the fundamental theorem of algebra, as well as the so-called "rule of signs" called after him. We present the section of Book Three that contains these results, in a translation based on The Geometry of René Descartes, translated from the French and Latin by D. E. Smith and M. L. Latham (Open Court, Chicago, London, 1925; Dover, New York, 1954), 159–163, 175; this edition also has the original French text.In the translation we have left the term "dimension" for our degree, "true" for positive roots and "false" for negative ones. The terms "real" and "imaginary" are also in Descartes’s text; the use of these terms in our modern sense begins here.Note that Descartes’s coefficients have special numerical values, so that he ignores Viète’s logistica speciosa as well as his own way of denoting constants by letters But he introduces x, y for the unknown and the variables and this has become standard practice.By going from Cardan via Viète to Descartes we have bypassed some important writers on algebra, among them Rafael Bombelli (c. 1530–after 1572) and Simon Stevin (1546–1620). Bombelli introduced operations with imaginary numbers in connection with the solution of the cubic equation. Stevin’s algebra, contained in his Arithmétique (Leiden, 1585), can be studied in an English paraphrase in the Principal works, IIB (Swets-Zeitlinger, Amsterdam, 1958).Bombelli and Stevin replaced the old notation for the powers of the unknown, by which each power was expressed by its own symbol (a method that had been used by Diophantus), by a notation which used numbers to express exponents, so that we find in Stevin for x, or a, etc., [figure table] for x2, or a2, etc., [figure table] for x0, or a0, etc. Another algebraist, William Oughtred (1574–1660), a Cambridge-educated English minister, author of Clavis mathematicae (London, 1631), has been credited with the introduction or improvement of mathematical symbols, such as the use of × for multiplication, and the introduction of [figure table] for proportion. See F. Cajori, William Oughtred (Open Court, Chicago, 1916), who also gives a sample of his work on equations in English translation. Those interested in the history of mathematical notations should consult F. Cajori, History of mathematical notations (2 vols.; Open Court, Chicago, 1928, 1929). On Oughtred’s older contemporary Harriot see note 1, and J. A. Lohne, "Thomas Harriot als Mathematiker," Centaurus 11 (1965), 19–45.

Mathematics

8 DESCARTES.

Theory of Equations

Name: A Source Book in Mathematics, 1200-1800 - Theory of Equations
Author: René Descartes

It is necessary that I make some general statements concerning the nature of equations, that is, of sums composed of several terms, in part known, in part unknown, of which some are equal to the others, or rather, all of which considered together are equal to zero, because this is often the best way to consider them.

Know then that in every equation there are as many distinct roots, that is, values of the unknown quantity, as is the number of dimensions of the unknown quantity.²

Suppose, for example,

³ and again,

Multiplying together the two equations

and

we have

This is an equation in which x has the value 2 and at the same time x has the value 3. If we next make

and multiply this by

we have

another equation, in which x, having three dimensions, has also three values, namely, 2, 3, and 4.

It often happens, however, that some of the roots are false, or less than nothing. Thus, if we suppose x to represent the defect⁴ of a quantity 5, we have

which, multiplied by

gives

as an equation having four roots, namely three true roots, 2, 3, and 4, and one false root, 5.

It is evident from this that the sum of an equation containing several roots is always divisible by a binomial consisting of the unknown quantity diminished by the value of one of the true roots, or plus the value of one of the false roots. In this way, the dimension of an equation can be lowered.

On the other hand, if the sum of an equation is not divisible by a binomial consisting of the unknown quantity plus or minus some other quantity, then this latter quantity is not a root of the equation. Thus the last equation

is divisible by

and

but is not divisible by x plus or minus any other quantity, which shows that the equation can have only the four roots, 2, 3, 4, and 5.

We can determine from this also the number of true and false roots that any equation can have, as follows: An equation can have as many true roots as it contains changes of sign, from + to - or from - to +; and as many false roots as the number of times two + signs or two - signs are found in succession.⁵ Thus, in the last equation, since +x⁴ is followed by -4x³, giving a change of sign from + to -, and - 19xx is followed by + 106x and + 106x by - 120, giving two more changes, we know there are three true roots; and since -4x³ is followed by - 19xx there is one false root.

It is also easy to transform an equation so that all the roots that were false shall become true roots, and all those that were true shall become false. This is done by changing the sign of the second, fourth, sixth, and all even terms, leaving unchanged the signs of the first, third, fifth, and other odd terms. Thus, if instead of

we write

we get an equation having only one true root, 5, and three false roots, 2, 3, and 4.

If the roots of an equation are unknown and it be desired to increase or diminish each of these roots by some known number, we must substitute for the unknown quantity another quantity greater or less by the given number.⁶ Thus, if it be desired to increase by 3 the value of each root of the equation

put y in the place of x, and let y exceed x by 3, so that

Then for xx put the square of

for x³ put its cube,

and for x⁴ put its fourth power [quarré de quarré], or

Substituting these values in the above equation, and combining, we have

whose true root is now 8 instead of 5, since it has been increased by 3.

If, on the other hand, it is desired to diminish by 3 the roots of the same equation, we must make

and

and so on, so that instead of

we have

It should be observed that increasing the true roots of an equation diminishes the false roots by the same amount; and on the contrary diminishing the true roots increases the false roots; while diminishing either a true or a false root by a quantity equal to it makes the root zero; and diminishing it by a quantity greater than the root renders a true root false or a false root true. Thus by increasing the true root 5 by 3, we diminish each of the false roots, so that the root previously 4 is now only 1, the root previously 3 is zero, and the root previously 2 is now a true root, equal to 1, since

This explains why the equation

has only three roots, two of them, 1 and 8, being true roots, and the third, also 1, being false; while the other equation

has only one true root, 2, since

and three false roots, 5, 6, and 7.

¹ As we have said, Descartes’s relation to Viète is not clear. The same holds for his relation to the English mathematician and astronomer Thomas Harriot (1560–1621), also known for his description of newly named Virginia (1588). Harriot wrote (probably about 1610) the influential Artis analyticae praxis (London, 1631), with a theory of equations which in several aspects anticipates Descartes’s treatment. He wrote

for our

(see Selection I.2) and used > and in our present sense: Signum majoritatis ut

significat a majorem quam b. John Wallis (1616–1703), in his Treatise of algebra (Oxford, 1685), claimed that Descartes borrowed heavily from Harriot: "Hariot hath laid the foundations on which Des Cartes (though without naming him) hath built the greatest part (if not the whole) of his Algebra or Geometry." Wallis missed the point, but his statement foreshadows the tension between English and Continental mathematicians typified in the later Newton-Leibniz priority struggle. See J. F. Scott, The mathematical work of John Wallis (Taylor and Francis, London, 1938), chap. IX.

² This is Descartes’s formulation of the fundamental theorem of algebra. For its further, more exact, formulation and proof, see Selections II.6 on Girard, II.10 on Euler, and II.12 on Gauss.

³ Descartes writes: "x equal to 2 or

equal to nothing, and again

He does not use the sign =, already introduced by Recorde and Harriot; see note 1.

⁴ A defect [défaut] is the negative of a positive number; thus -5 is the defect of 5, that is, the remainder when 5 is subtracted from zero.

⁵ This is the sign rule as stated by Descartes. It was formulated in a more precise manner by Isaac Newton in his Arithmetica universalis (Cambridge, 1707) and by C. F. Gauss in "Beweis eines algebraischen Lehrsatzes," Crelle’s Journal [figure table]

die reine und angewandte Mathematik 3 (1828), 1–4; Werke, III, 65–70. We now can express it in the following way: If

is an equation of degree n with real coefficients, where

then the number of positive roots is equal to or an even number less than the number of variations in the signs of successive terms. Multiple roots have to be counted in accordance with their multiplicity, and zero is not a positive root.

⁶ This change of variable of an equation by means of substitution of the type

is not new, and is, as we observed in Selection II.4, note 8, one of the substitutions used by Cardan in his Ars magna (1545). Descartes here shows its use in the search for positive and negative roots. The notation, as elsewhere in Descartes’s writings, strikes us as quite modern.

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Chicago: René Descartes, "Theory of Equations," A Source Book in Mathematics, 1200-1800, ed. C. Adam and P. Tannery in A Source Book in Mathematics, 1200-1800, ed. D. J. Struik (Princeton: Princeton University Press, 1969, 1986), 89–93. Original Sources, accessed April 24, 2024, http://originalsources.com/Document.aspx?DocID=WBGLY8NNNPLUNIG.

MLA: Descartes, René. "Theory of Equations." A Source Book in Mathematics, 1200-1800, edited by C. Adam and P. Tannery, Vol. VI, in A Source Book in Mathematics, 1200-1800, edited by D. J. Struik, Princeton, Princeton University Press, 1969, 1986, pp. 89–93. Original Sources. 24 Apr. 2024. http://originalsources.com/Document.aspx?DocID=WBGLY8NNNPLUNIG.

Harvard: Descartes, R, 'Theory of Equations' in A Source Book in Mathematics, 1200-1800, ed. . cited in 1969, 1986, A Source Book in Mathematics, 1200-1800, ed. , Princeton University Press, Princeton, pp.89–93. Original Sources, retrieved 24 April 2024, from http://originalsources.com/Document.aspx?DocID=WBGLY8NNNPLUNIG.