François Viète: Genius Codebreaker Who Transformed Algebra

Gabrielle Birchak/ November 19, 2024/ Early Modern History, Middle Ages, Modern History, Post Classical/ 0 comments

Ten­sions are ris­ing between two pow­er­ful nations, each try­ing to out­ma­neu­ver the oth­er in a high-stakes game of intel­li­gence. In the mid­dle of this stand­off, a cod­ed mes­sage is inter­cept­ed. It’s a mes­sage that could change every­thing. The usu­al cryp­tog­ra­phers can’t crack it; the code seems unbreak­able. Enter a lawyer and a self-taught crypt­an­a­lyst with an unmatched gift for spot­ting hid­den pat­terns, some­one who’s built his math­e­mat­i­cal skills on sheer curios­i­ty and relent­less prac­tice. As he starts decod­ing, line by line, some­thing aston­ish­ing comes to light. This rev­e­la­tion could shift the entire bal­ance of power.

In 1589, France was gripped by unrest. Protes­tant King Hen­ry IV faced con­stant threats from Catholic Spain, led by King Philip II, who sup­port­ed French Catholic forces, height­en­ing ten­sions. Amidst this tur­moil, a cod­ed Span­ish let­ter was inter­cept­ed, but its cipher baf­fled French cryptographers.

Enter François Viète, a renowned math­e­mati­cian. When he took on the cipher, he noticed recur­ring pat­terns, apply­ing alge­bra­ic tech­niques to unrav­el the mes­sage. His efforts revealed a cor­re­spon­dence between Philip II and his gen­er­als, detail­ing Span­ish plans to back French Catholic fac­tions against Hen­ry IV. Viète informed French offi­cials, pro­vid­ing them with crit­i­cal intel­li­gence to counter Spain’s moves.

In a bold step, Viète pub­lished his deci­pher­ing work with the King’s con­sent. He even shared intel­li­gence with the Catholic League’s leader, Duke of Mayenne, real­iz­ing that Spain’s sup­port for French Catholics aimed to divide France. Viète knew secre­cy was cru­cial, as the Span­ish would devel­op more com­plex ciphers after their code was cracked. Soon, he was over­whelmed with dis­patch­es, enlist­ing sec­re­taries to man­age the influx.

Philip II was furi­ous and accused Viète of sor­cery, even appeal­ing to the Pope for excom­mu­ni­ca­tion. Yet, Viète’s achieve­ment wasn’t magic—it was an ear­ly tri­umph of crypt­analy­sis, demon­strat­ing the pow­er of ana­lyt­i­cal thinking.

Viète’s suc­cess marked a turn­ing point in the French Wars of Reli­gion, weak­en­ing Spain’s influ­ence. The intel­li­gence he gath­ered hin­dered Span­ish efforts to con­trol France and helped lead to the Treaty of Vervins in 1598, end­ing open hos­til­i­ties. This con­flict, fueled by deep Catholic-Protes­tant ten­sions, began in 1562 and per­sist­ed through decades of vio­lence and for­eign inter­ven­tion. Hen­ry IV’s con­ver­sion to Catholi­cism in 1593 and the 1598 Edict of Nantes, grant­i­ng Huguenots reli­gious free­dom, tem­porar­i­ly eased ten­sions. How­ev­er, the Edict’s revo­ca­tion in 1685 reignit­ed per­se­cu­tion until the sec­u­lar­ism of the French Rev­o­lu­tion estab­lished last­ing reli­gious tol­er­ance by the 18th century.

Viète’s work showed that intel­lect could shape the fate of nations, lay­ing the ground­work for mod­ern cryp­tog­ra­phy and a new kind of warfare.

François Viète

François Viète, born in 1540 in Fonte­nay-le-Comte, France, was a math­e­mati­cian and lawyer cel­e­brat­ed for his pio­neer­ing con­tri­bu­tions to alge­bra and crypt­analy­sis. He stud­ied law at the Uni­ver­si­ty of Poitiers, quick­ly estab­lish­ing a rep­u­ta­tion for his ana­lyt­i­cal skills, which led him to serve as an advi­sor in the French roy­al court under both King Hen­ry III and King Hen­ry IV. Despite his suc­cess in law, Viète’s pri­ma­ry pas­sion was math­e­mat­ics, where he intro­duced rev­o­lu­tion­ary sym­bol­ic nota­tion that laid the foun­da­tion for mod­ern alge­bra. His work, bridg­ing law, math­e­mat­ics, and cryp­tog­ra­phy, show­cas­es him as one of the Renaissance’s most inter­dis­ci­pli­nary minds.

In addi­tion to his con­tri­bu­tions to alge­bra, Viète was known for his cryp­to­graph­ic skills, notably dur­ing a time of height­ened ten­sions between France and Spain. His sys­tem­at­ic approach to break­ing the Span­ish ciphers demon­strat­ed his alge­bra­ic insight and pat­tern-recog­ni­tion skills. For exam­ple, he ana­lyzed redun­dant sym­bols in cod­ed let­ters, iden­ti­fied repeat­ing pat­terns, and uti­lized fre­quen­cy analy­sis, a sta­tis­ti­cal tech­nique that involves count­ing repeat­ed ele­ments to uncov­er hid­den infor­ma­tion. His care­ful analy­sis extend­ed to tri­grams and digrams, pat­terns of three and two let­ters, help­ing him iden­ti­fy mean­ing­ful sym­bols and decode the mes­sages. These meth­ods, although straight­for­ward, were inno­v­a­tive for their time and empha­sized Viète’s ana­lyt­i­cal prowess, as he essen­tial­ly laid the ground­work for mod­ern cryp­to­graph­ic analysis.

Cryptographic Methods

Viète’s Approach to Deciphering

In the 1997 arti­cle titled “Secrets, Sym­bols, and Sys­tems” the author Peter Pesic describes the meth­ods that Viète wrote before he passed away.

Viète’s method of break­ing down the code used his alge­bra­ic insights to look for pat­terns and fre­quen­cies in the sym­bols. He sub­sti­tut­ed let­ter com­bi­na­tions and iden­ti­fied repeat­ed pat­terns, a pre­cur­sor to mod­ern cryp­to­graph­ic analy­sis. This math­e­mat­i­cal approach, cou­pled with his patience and log­i­cal skills, grad­u­al­ly cracked the code’s structure.

Redun­dant Symbols

First, Viète looked for redun­dant sym­bols. As in the case of the Span­ish let­ter, he com­pared dif­fer­ent copies of a let­ter sent by an ambas­sador. By com­par­ing the two, he was able to find redun­dant symbols.

Then, he looked at the val­ues that were pre­sent­ed in the doc­u­ment. These are known as “essen­tial num­bers.” They rep­re­sent the num­ber and are not cod­ed sym­bols. How­ev­er, he could then look at the word­ing around those val­ues. Pesic explains that these essen­tial num­bers allowed crypt­an­a­lysts to find the words between them. He gives the exam­ple “12 m 38 13,” where m38 is flagged by the num­bers 12 and 13. As a result, Viète real­ized that the code “m 38” rep­re­sent­ed the Span­ish word “ou,” which means “or.” This would change the phrase to read “12 or 13.”

Con­ven­tion­al Phrases

Viète then looked at con­ven­tion­al phras­es that were used in Spain and often used in their lit­er­a­ture, such as “copia de” and “memo­r­i­al y instruc­tion.” These phras­es usu­al­ly indi­cat­ed that a code would be near­by that he could decipher.

Fre­quen­cy Analysis

Viète then uti­lized fre­quen­cy analy­sis. In sta­tis­tics, fre­quen­cy analy­sis is a method that counts how many times events or val­ues occur with­in a data set. Iden­ti­fy­ing these pat­terns and trends allows the crypt­an­a­lyst to deter­mine whether the fre­quen­cy of those phras­es or let­ters is used more than they nor­mal­ly would. Usu­al­ly this would then indi­cate that there is code with­in those patterns.

Tri­grams

Viète real­ized that the suc­ces­sion of three con­so­nants rarely occurs in French, Span­ish, and Ital­ian. Thus, if you had three let­ters and two were con­so­nants, the third let­ter would be a vow­el. He would then find a group of mutu­al­ly exclu­sive tri­ads from the text, where each would con­tain a vow­el, and those vow­els would lead to anoth­er code when extract­ed. This was what Viète coined his “infal­li­ble rule.”

Digrams

This process was not mutu­al­ly exclu­sive to three let­ters. He also looked for the suc­ces­sion of two repeat­ed let­ters. These are known as digrams or dyads.

All these meth­ods uti­lize a monoal­pha­bet­ic cipher, a sim­ple cipher involv­ing let­ters and num­bers. How­ev­er, ciphers can also include sym­bols or use num­bers or let­ters as symbols.

Thus, Viète’s method wasn’t found­ed on genius, intu­ition, or insight. It was a sim­ple, sys­tem­at­ic process. Viète had cre­at­ed a straight­for­ward crypt­analy­sis process that uti­lized the same prac­tices repeatedly.

Viète’s math­e­mat­i­cal think­ing and his con­tri­bu­tions to cryp­tog­ra­phy are intrin­si­cal­ly linked through the prin­ci­ples of analy­sis, abstrac­tion, and sys­tem­at­ic prob­lem-solv­ing. The par­al­lels between these two fields high­light why Viète was unique­ly qual­i­fied to tack­le cryp­to­graph­ic chal­lenges dur­ing his time. His meth­ods also high­light the pow­er of inter­dis­ci­pli­nary think­ing. He was a lawyer and legal advi­sor. Yet the process of ana­lyt­i­cal think­ing and his abil­i­ty to break down com­plex prob­lems into sim­pler com­po­nents allowed him to approach the Span­ish cipher with a sys­tem­at­ic mindset.

And for attor­neys, or any­body who has watched Law and Order, The Prac­tice, Suits, or any oth­er legal show, you would know that attor­neys look for pat­terns and struc­tures in behav­ior and out­comes. They would then use that analy­sis to defend their clients. Like­wise, math­e­mat­ics is often about iden­ti­fy­ing pat­terns and struc­tures, which lends itself to cryp­tog­ra­phy. Viète’s keen eye for pat­terns trans­lat­ed seam­less­ly to his work in deci­pher­ing codes.

Fur­ther­more, log­i­cal rea­son­ing is foun­da­tion­al to law and alge­bra. Viète’s method­i­cal approach to alge­bra involved form­ing hypothe­ses, test­ing them, and refin­ing his strate­gies based on results. This log­i­cal rea­son­ing is equal­ly essen­tial in crypt­analy­sis, where one must eval­u­ate dif­fer­ent sub­sti­tu­tion meth­ods and val­i­date their effec­tive­ness against the encod­ed message.

Viete’s Alge­bra­ic Methods

Before François Viète’s con­tri­bu­tions, alge­bra lacked a stan­dard­ized sym­bol­ic nota­tion, so math­e­mati­cians often used words or lengthy phras­es to describe oper­a­tions and rela­tion­ships as far back as 1000 BCE, when rhetor­i­cal alge­bra was imple­ment­ed. Known quan­ti­ties were typ­i­cal­ly expressed through phras­es, and unknowns weren’t con­sis­tent­ly rep­re­sent­ed with sym­bols. By 200 CE, syn­co­pat­ed alge­bra was imple­ment­ed, as pre­sent­ed by Dio­phan­tus and Brah­magup­ta. In syn­co­pat­ed alge­bra, sym­bol­ism was used, but it didn’t include all the appli­ca­tions of sym­bol­ic alge­bra. Solu­tions relied on spe­cif­ic rules for each type of prob­lem, often tied to indi­vid­ual cas­es. For exam­ple, sub­trac­tion could only be used on one side of an equa­tion in syn­co­pat­ed alge­bra. These approach­es made it chal­leng­ing to gen­er­al­ize or sys­tem­at­i­cal­ly manip­u­late equations.

His ground­break­ing work pro­vid­ed alge­bra with a sol­id foun­da­tion mark­ing the end of medieval algebra’s pro­ce­dur­al roots from schol­ars like Al-Khwariz­mi. By cre­at­ing sym­bol­ic alge­bra, Viète claimed that this inno­v­a­tive approach could solve all prob­lems. This is a bold but true asser­tion known as nul­lum non prob­le­ma sol­vere. His approach laid the ground­work for mod­ern alge­bra­ic meth­ods and sym­bol­ized a shift toward uni­ver­sal prob­lem-solv­ing in mathematics.

In his ded­i­ca­tion of The Ana­lyt­i­cal Art to Cather­ine de Parthenay, Viète intro­duced his rev­o­lu­tion­ary approach, refer­ring to it as new. He believed alge­bra had been degrad­ed by ear­li­er inter­pre­ta­tions and sought to restore it through a new vocab­u­lary, free from unnec­es­sary tech­ni­cal jar­gon. His work empha­sized clar­i­ty and order, pre­sent­ing his ideas as the first steps in what would even­tu­al­ly be per­fect­ed over gen­er­a­tions. Despite his inno­va­tions, Viète lacked a few mod­ern nota­tions, such as mul­ti­pli­ca­tion sym­bols and the equal sign intro­duced by William Oughtred and Robert Recorde, respectively.

Viète’s sym­bol­ic choic­es were dis­tinc­tive, using con­so­nants for para­me­ters and vow­els for unknowns, fol­low­ing the styl­is­tic prece­dent of Petrus Ramus, a con­tem­po­rary who used sim­i­lar nota­tion in geom­e­try. This nota­tion, how­ev­er, did not gain pop­u­lar­i­ty with lat­er math­e­mati­cians, includ­ing Descartes, who chose oth­er meth­ods. Addi­tion­al­ly, Viète con­tin­ued the clas­si­cal prac­tice of treat­ing quan­ti­ties like lines and sur­faces rather than abstract num­bers. This choice com­pli­cat­ed his work and required him to ver­i­fy results geometrically.

Viète’s work, The Ana­lyt­i­cal Art, pub­lished in 1591, intro­duced the use of let­ters as sym­bols for known and unknown quan­ti­ties, rev­o­lu­tion­iz­ing alge­bra. Before Viète, math­e­mati­cians relied heav­i­ly on ver­bose lan­guage and numer­i­cal meth­ods with­out a sys­tem­at­ic nota­tion for oper­a­tions. By allow­ing gen­er­al for­mu­las and abstract manip­u­la­tion, Viète’s nota­tion trans­formed alge­bra into a uni­ver­sal lan­guage for solv­ing com­plex prob­lems. His method pro­vid­ed a frame­work that allowed for pre­cise sym­bol­ic manip­u­la­tion and was a foun­da­tion for what we now call sym­bol­ic alge­bra. This leap in math­e­mat­i­cal thought marked a shift away from medieval prac­tices and set a new stan­dard in prob­lem-solv­ing approaches.

The lega­cy of Viète’s alge­bra­ic meth­ods went beyond nota­tion. He intro­duced the bino­mi­al for­mu­la and devel­oped what are now called Viète’s for­mu­las, which relate the roots of poly­no­mi­als to their coef­fi­cients, a key devel­op­ment in alge­bra­ic the­o­ry that sim­pli­fied solv­ing poly­no­mi­al equa­tions. His inno­va­tions in trigonom­e­try, par­tic­u­lar­ly Viète’s for­mu­las, allowed math­e­mati­cians to extract infor­ma­tion about poly­no­mi­al roots with­out ful­ly solv­ing them, a pre­cur­sor to lat­er tech­niques in poly­no­mi­al analy­sis by math­e­mati­cians like Blaise Pas­cal and Isaac Newton.

Viète’s use of let­ters for knowns and unknowns allowed for broad­er, more abstract manip­u­la­tion. This set the stage for alge­bra as a gen­er­al­ized and sys­tem­at­ic branch of math­e­mat­ics known as sym­bol­ic algebra.

Viète’s for­mu­las are a bit like reverse-engi­neer­ing a recipe, where one can deduce ingre­di­ents (or in this case, solu­tions) based on the observed out­come. For my non­math audi­ence, Vièta’s for­mu­las are a set of equa­tions that, when eval­u­at­ed through his meth­ods, allow you to find data about the roots with­out ful­ly solv­ing the equation.

For exam­ple, imag­ine that you have a recipe that uses two pri­ma­ry fla­vors when they are com­bined. Let’s say the ingre­di­ents are papri­ka and gar­lic, two of my favorites. When you taste them, you don’t know how much gar­lic or papri­ka went into the meal. Still, you can taste garlic’s bit­ing taste and the paprika’s sweet, pep­pery taste. Vieta’s for­mu­las are like a way to reverse-engi­neer that recipe. So when you see a poly­no­mi­al, Vieta’s for­mu­las help us break down how much of each ingre­di­ent, or in this case, each solu­tion, con­tributes to the whole equa­tion just by look­ing at the num­bers in the equa­tion. So, instead of tast­ing papri­ka and gar­lic, we’re look­ing at the fla­vors of math, which are the sum and the prod­uct of the roots, oth­er­wise known as solu­tions. Vieta’s for­mu­las help us fig­ure out the hid­den ingre­di­ents with­out direct­ly solv­ing the equation.

For my math audi­ence, in most sim­plis­tic terms, Vieta’s for­mu­las are a set of equa­tions that relate the coef­fi­cients of a poly­no­mi­al to sums and prod­ucts of its roots. There is so much more to explore with Vieta’s for­mu­las. You can explore the inter­webs for so many dif­fer­ent expla­na­tions. How­ev­er, one of my favorite expla­na­tions is on the YouTube chan­nel Prime New­tons. Be sure to check it out.

Viète’s con­tri­bu­tions even extend­ed into alge­bra­ic geom­e­try, as he explored ways to con­struct alge­bra­ic expres­sions using only a ruler and com­pass. In his work A canon­i­cal review of geo­met­ri­cal effects, one of his for­ward-think­ing ideas was the prin­ci­ple of homo­gene­ity, requir­ing that all quan­ti­ties in an equa­tion share the same dimen­sion, which would lat­er be a cor­ner­stone of dimen­sion­al analy­sis. His meth­ods in sym­bol­ic cal­cu­la­tion, espe­cial­ly his three-part process in solv­ing equa­tions (zetet­ic, poris­tic, and exegetic), stream­lined com­plex prob­lem-solv­ing by break­ing equa­tions into sys­tem­at­ic phases—a method that would become an endur­ing tool for future mathematicians.

Viète’s influ­ence wasn’t lim­it­ed to France. His rep­u­ta­tion grew when he famous­ly solved a com­plex math­e­mat­i­cal prob­lem posed by the Dutch math­e­mati­cian Adri­aan van Roomen. At the encour­age­ment of King Hen­ry IV, Viète pro­vid­ed an ele­gant solu­tion, impress­ing van Roomen, who trav­eled to France to study with him, ini­ti­at­ing a col­lab­o­ra­tive friend­ship that illus­trat­ed Viète’s wide-reach­ing impact on Euro­pean mathematics.

In the broad­er his­tor­i­cal con­text, Viète’s work marked a defin­i­tive end to the ad hoc rules and ver­bal descrip­tions that had dom­i­nat­ed alge­bra from medieval times. His sym­bol­ic approach pre­sent­ed alge­bra not only as a pro­ce­dur­al tool but as a struc­tured, log­i­cal sys­tem. By intro­duc­ing abstract rea­son­ing and empha­siz­ing sym­bols, Viète’s con­tri­bu­tions were foun­da­tion­al to alge­bra as we under­stand it today. His influ­ence was pro­found, paving the way for the alge­bra­ic meth­ods used in mod­ern math­e­mat­ics and cryp­tog­ra­phy, and his inter­dis­ci­pli­nary genius con­tin­ues to be cel­e­brat­ed cen­turies later.

Viète’s rig­or and imag­i­na­tive syn­the­sis of alge­bra­ic and geo­met­ric prin­ci­ples con­tin­ue to inspire math­e­mati­cians, crypt­an­a­lysts, and com­put­er sci­en­tists. For me, the sto­ry of Viète rep­re­sents the ulti­mate in inter­dis­ci­pli­nary think­ing and appli­ca­tions. So many schools raise our kids to be spe­cial­ized with­out even con­sid­er­ing the val­ue of embrac­ing mul­ti­ple branch­es of learning.

By embody­ing a spir­it of inquiry that bridges var­i­ous dis­ci­plines, his lega­cy reminds us how inno­v­a­tive approach­es can advance soci­ety, sci­ence, tech­nol­o­gy, and secu­ri­ty in our increas­ing­ly dig­i­tal world. Over 400 years, François Viète and his work remain a tes­ta­ment to the pow­er of abstract thought, sym­bol manip­u­la­tion, and rig­or­ous method­ol­o­gy that inspire the con­tin­u­ous pur­suit of math­e­mat­i­cal dis­cov­ery. Viète showed the world that you don’t even have to be a sci­en­tist or math­e­mati­cian to change the future of STEM.