ad aequationes algebraicae nova methodo resolvendas

This truly historic work * 'The Practise of the Analytic Art......' * helped to change the way mathematics would be done for ever: for it was an attempt to wean the contemporary mathematician from thinking along the age old lines of Greek Geometry to follow those set out by Vieta originally, in the fledgling art of Algebra. This was done by demonstrating the incredible convenience arising from establishing and solving equations using symbols rather than using the long-winded word equations of Vieta, who was nevertheless the true founder of algebra apart from the efforts of Diophantes. The book was published in 1631 in London, though the methods it expounded had been in use for some time. Considered to be the brainchild of Thomas Harriot, the *Praxis* was pieced together by his former associate Walter Warner some ten years after Harriot's untimely death. Harriot had set aside money in his will for Nathanial Torporley, another associate, to edit his work, but after 10 years there was no publication in sight, and the original benefactor, Henry Percy, the 8th Earl of Northumberland, was not long for this world; hence Harriot's other associates decided to do something, and the *Praxis* was the outcome. Warner was a mathematician in his own right, and at some time - though it is not clear exacly when - he was in contact with the European mathematicians, including Marat Mersenne, and one may presume Descartes himself, either directly or indirectly. Thus, Warner had a chapter on optics in Mersenne's * 'Cogitata' * of 1644. Though we are assured by the editor that the Praxis is based on Harriot's work, we should perhaps look on the work as indicative of the state of the art in the late 1620's, rather than a faithful reproduction of Harriot's ideas from some 20 or 30 years before. The *Praxis* is thus a sort of vademecum; only occasionally does the glittering genius of Thomas Harriot show through : but for this reason if none other, it is worth a read. The first part or * logistices speciosae * is mainly concerned with classifying quadratic, cubic, and quartic equations into canonical or standard forms according to the nature of their positive roots; the process of parodisation (removal of a power) is considered at length; inequalities from the coefficients are used to assert the number of positive roots; and equations with related roots are also considered, in Sections 1 to 6. The second part or* logistices numerosae *is taken up with the numerical solution of such equations following Vieta, using the rules established in the first part.

The interested reader can do little better than to consult the works of Shirley on Harriot for an unbiased account of Harriot's life and times. Harriot produced much that was new and original, but published none of it; thus, the arrival of algebra on the scene is rather remeniscent of what happened a little later with the arrival of the calculus, which seem to owe much to Isaac Barrow initially: who instilled the main ideas into his protege Newton (who was not originally a mathematician). For apart from Barrow's * Lectiones Geometriae*, (see Child's translation), again there was no publication for some time. Unfortunately, there was no handy Newton to whom Harriot could pass on his ideas. Such 'closet mathematics' led to bitter priority disputes which usually had a nationalistic flavour. Harriot essentially took Vieta's work on algebra a step further, by introducing symbols for everything in an equation (apart from actual numbers), with the vowel 'a' for the unknown. One may recall that Harriot had done an analogous sort of thing in his work on noting down the sounds of Red Indian speech for use in a dictionary of tribal languages during his American travels (he was also an accomplished classical linguist). Is it too far-fetched to conjecture that Harriot thought in terms of symbols from this experience? The process was established, however, and mathematicians to this very day continue to follow it with enthusiasm.

CONTENTS

Preface: In which a brief history of the subject from earliest times to Vieta is given. Link to this document by clicking here. You can do likewise to access any section; use the browser 'Back' arrow to return to this screen.

Definitions.

An introduction to the terminology and methods of algebra; mainly due to Vieta.

First Section.

The *Logistices Speciosae*, in which the four kinds of operation are made clear from examples: these show addition, subtraction, multiplication, and division using symbols rather than numbers. Definition of the greater than and less than signs are given. Examples show how simple equations can be manipulated.

Second Section.

In which the derivation of the equations of the first canonical form are established through multiplication together of factors of the form * a - b, a - c, *, etc. The expansions are given for quadratic, cubic, and bi-quadratic equations in terms of powers of the unknown *a* with coefficients expressed as sums and differences of the roots *b, c, d*, etc.

Third Section A

In which the equations of the second canonical form are deduced or reduced from the first by means of some relation established between the roots. This results in the removal of some power, with the remaining roots of the equation remaining unchanged.

Third Section B.

More involved calculations performed in the reduction of equations to simpler forms. Final summary of forms investigated.

Fourth Section

In which the roots of the equations of the first canonical form, as of the reduced equations, are established. Thus, the *Praxis *assumes the positive roots, and shows the canonical equation they satisfy.

Fifth Section.

In which the number of the roots of homogeneous common equations are determined from their equivalent equipollant canonical forms. This involves the use of inequalities formed from the coefficients.

Sixth Section

In which reduced equations are further considered. Doubling and trebling of the roots of quadratic and cubic equations is demonstrated; changes are made to reduced cubic equations by changing the binomial factors, thus changing the affected terms. The solutions of some appropriately reduced cubics are shown, related to Vieta's method, and changes to the equipollant forms of quartic equations are finally shown.

Exegetice Numerosa Part I.

In which quadratic equations are solved numerically for positive roots by the method of Vieta.
A number of special cases are considered.

Exegetice Numerosa Part II.

In which cubic equations are solved numerically for positive roots by the method of Vieta.
A number of special cases are considered; the method mirrors that for quadratics.

Exegetice Numerosa Part III.

In which biquadratic and quintic equations are solved numerically as above, and the analysis is extended to irrational roots.

Exegetice Numerosa Part IV.

A directory of the canons is presented : 4 for the quadratic; 14 for the cubic ; and 46 for the quartic.

Ian Bruce. August 2006 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for personal or educational use.