My eponymic contribution to Sexagesimal math

Gabriellebirchak/ January 26, 2021/ Ancient History, Modern History, Post Classical

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Last month, as I cov­ered a great deal of mate­r­i­al on Tes­la, I found a cool mul­ti­pli­ca­tion cir­cle called Tesla’s Mul­ti­pli­ca­tion Map. This mul­ti­pli­ca­tion map is a spi­ral of num­bers, with its inner­most num­bers rep­re­sent­ing a clock’s lay­out. Then, the next num­ber out­side of that first inner cir­cle begins with 13, spi­ral­ing around. Even­tu­al­ly, you have a large spi­ral of num­bers that can go as high as you want, as long as each spi­ral of num­bers is a mul­ti­ple of 12. 

The beau­ti­ful thing about this spi­ral is that you can draw dif­fer­ent shapes when you con­nect the num­bers based on mul­ti­ples of 2, 3, 4, 5, 6, and on and on. What is also fan­tas­tic about this spi­ral is that it unique­ly teach­es mul­ti­pli­ca­tion so that stu­dents can have an artis­tic visu­al of multiplication.

I want­ed to try it for myself, so I made my own mul­ti­pli­ca­tion map build­ing a spi­ral up the val­ue of 60. Then I drew out sev­er­al shapes con­nect­ing num­bers divis­i­ble by 2, 3, 4, 5, and 6. Each val­ue gave me a dif­fer­ent design.

A teacher in Ari­zona by the name of Joey Grether attrib­uted this map to Tes­la. Grether inten­tion­al­ly named it Tesla’s Mul­ti­pli­ca­tion Map because the name Grether’s Mul­ti­pli­ca­tion Map could not seem to gain trac­tion. He designed it to look like an old doc­u­ment and titled it “N. Tes­la – Map to Mul­ti­pli­ca­tion.” Then Grether fab­ri­cat­ed a sto­ry that a local artist by the name of Abe Zuc­ca found it at an antique shop in Phoenix. Accord­ing to Grether’s sto­ry, Zuc­ca shared it with Grether, who “had a few break­throughs,” stat­ing that the map “offers a com­pre­hen­sive visu­al under­stand­ing of how all num­bers are self-orga­nized into 12 posi­tions of com­positabil­i­ty.”[i]

This mis­at­tri­bu­tion is called an eponymy. Eponymy is a the­o­ry or an idea is named after the wrong per­son, which is usu­al­ly the per­son with the most noto­ri­ety. In Grether’s case, he inten­tion­al­ly named it after Tes­la to gain inter­net trac­tion for this map. Eponymy comes from the evi­dence that shows how cel­e­brat­ed sci­en­tists get more cred­it than unknown researchers do. Amer­i­can soci­ol­o­gist Robert Mer­ton called this the Matthew effect, which he ref­er­enced from the Bible verse Matthew 25:29, which states, “For who­ev­er has will be giv­en more, and they will have an abun­dance. Who­ev­er does not have, even what they have will be tak­en from them.” 

The cor­re­la­tion between eponymy and Merton’s Matthew Effect are ref­er­enced in Stigler’s Law of Eponymy. Dr. Stephen Stigler named Mer­ton as the dis­cov­er­er of “Stigler’s Law of Eponymy” in his paper of the same name pub­lished in 1980. This paper was a trib­ute to Mer­ton from Stigler when Mer­ton retired in 1979. His paper’s title was a clever joke on Stigler’s behalf based on Merton’s theory.

The female term of the Matthew effect is called the Matil­da effect. How­ev­er, the name Matil­da Effect did not come from a bible verse. Instead, the Matil­da Effect was named after Matil­da Joslyn Gage, who was born in 1826 and was a suf­frag­ist, and activist, and orga­niz­er of the Women’s Nation­al Lib­er­al Union. For my long time lis­ten­ers, you might recall my Octo­ber 2019 pod­cast on the Matil­da Effect, which pro­vid­ed a large list of female sci­en­tists that pres­ti­gious orga­ni­za­tions inten­tion­al­ly over­looked. One such sci­en­tist includes Dr. Vera Rubin, whose work on dark mat­ter earned Dr. James Pee­bles the Nobel Prize in Physics.

Anoth­er not­ed female sci­en­tist includes Dr. Marie Curie. She won her first Nobel in 1903, but only after her hus­band, Pierre, advised the Nobel com­mit­tee that Marie had a sig­nif­i­cant role in her dis­cov­er­ies of radioactivity.

Then there was Dr. Ros­alind Franklin, a British biol­o­gist and DNA Pio­neer. Though she dis­cov­ered the struc­ture of DNA. she nev­er won the Nobel for her work. Instead, the male mem­bers of her team won the Nobel in Phys­i­ol­o­gy or Medicine. 

In ancient his­to­ry, one of the first known mis­at­tri­bu­tions comes from 300 CE when the math­e­mati­cian Pap­pus ref­er­enced anoth­er math­e­mati­cian by the name of Pan­dro­sion. His­tor­i­cal writ­ers had iden­ti­fied Pan­dro­sion as a man. How­ev­er, Pan­dro­sion was one of the first known female math professors. 

Oth­er ear­li­er known forms of eponymy include the Fibonac­ci sequence, which is 1,1,2,3,5,8,13,21,34,55, and on and on. This sequence is a set of num­bers first attrib­uted to Fibonac­ci by the math­e­mati­cian Édouard Lucas in the late 1800s. Lucas not­ed that each new num­ber is equal to the sum of the pre­vi­ous two num­bers. How­ev­er, this sequence was first pre­sent­ed in 200 BCE by an ancient Indi­an author Acharya Pin­gala.[ii]

Then we have the Has­se dia­gram, which is a math­e­mat­i­cal dia­gram used to rep­re­sent a finite par­tial­ly ordered set. This dia­gram is a two- or three-dimen­sion­al object, where a point con­nects each line. Each of those points rep­re­sents an ele­ment in the set that is par­tial­ly ordered. These dia­grams are attrib­uted to Hel­mut Has­se. How­ev­er, the math­e­mati­cian Hen­ri Gus­tav Vogt first used these dia­grams three years before Has­se was even born.[iii]

Then we have Newton’s first law of mechan­ics, which states, “A par­ti­cle in motion will con­tin­ue to move in the straight line, at a con­stant speed, unless that par­ti­cle is act­ed upon by an exter­nal force.” He pub­lished this first law, among oth­er laws, in his work Math­e­mat­i­cal Prin­ci­pals of Nat­ur­al Phi­los­o­phy in 1687. How­ev­er, Newton’s First Law of Motion is also known as Galileo’s Law of Iner­tia. Many decades ear­li­er, Galileo fig­ured out that a body of motion will remain in motion unless some­thing like fric­tion will cause it to rest.

Anoth­er eponymy includes the Bilin­s­ki dodec­a­he­dron, which is a 12-sided com­plex poly­he­dron with con­gru­ent rhom­bic faces. The Bilin­s­ki dodec­a­he­dron was named after Stanko Bilin­s­ki, a Croa­t­ian math­e­mati­cian who redis­cov­ered it in 1960.[iv] How­ev­er, in 1752, John Lodge Cow­ley, a car­tog­ra­ph­er, geol­o­gist, and math­e­mati­cian, orig­i­nal­ly pre­sent­ed this par­tic­u­lar 12-sided com­plex poly­he­dron in his book Geom­e­try Made Easy.

Final­ly, we have the Pythagore­an The­o­rem, which is attrib­uted to the Greek philoso­pher Pythago­ras. The Baby­lo­ni­ans had been using this the­o­rem 1,300 years before Pythago­ras was even born. The Baby­lo­ni­ans knew that the square of the hypotenuse is equal to the sum of the squares of the two sides. Addi­tion­al­ly, oth­er cul­tures used this the­o­rem as well. 

The Plimp­ton-322, from 1800 BCE, is a clay tablet with num­bers etched into four columns and fif­teen rows. What may look like a ran­dom set of Semit­ic cuneiform etch­es is real­ly a list of num­bers rep­re­sent­ing Pythagore­an triples, which are a list of num­bers that rep­re­sent the length, width, and diam­e­ter of a right tri­an­gle. When deduced, all these val­ues show that the square of the hypotenuse is equal to the sums of the squares of each side. This has some of the old­est evi­dence of their use of base 60. 

I think it is fas­ci­nat­ing that the mul­ti­pli­ca­tion map, the spi­ral, uses foun­da­tions in Baby­lon­ian math­e­mat­ics and uses com­po­nents of base 60.

That is why I built my Insta­gram mul­ti­pli­ca­tion map up to the val­ue of 60 and a print­out for you up to the val­ue of 120, which is 2 times 60. For those who have not heard my pre­vi­ous pod­cast where I talked about base 60, base 60 is a sex­a­ges­i­mal numer­al sys­tem. What that means is that instead of using ten as a base for all math­e­mat­ics, ear­ly math­e­mati­cians used 60. For us today, we break down our num­bers based on units of 10. We learn in ele­men­tary school how to count to 10 using our fin­gers. When we do addi­tion, we add in base 10 car­ry­ing the val­ues over once we exceed 10.

How­ev­er, in ancient Baby­lon, when they added in base 60, they car­ried the val­ues over when they exceed­ed 60. Con­duct­ing math in base 60 last­ed for hun­dreds of years, into the sev­enth cen­tu­ry of our cur­rent era. It may seem like this val­ue of 60 is hard to com­pre­hend because it is such a large num­ber, but they man­aged to make it work.

The Sume­ri­ans and Baby­lo­ni­ans also count­ed to 60 on their fin­gers. Look­ing at your right-hand point­er fin­ger, you will see that your fin­ger has three seg­ments between each joint. Each of those seg­ments on the fin­ger is called a pha­lange. When you use your thumb to touch each pha­lange on your right hand, start­ing with your point­er fin­ger, you will be able to count up to 12. Now, as we con­tin­ue to count, we use the left-hand dig­its to count for every val­ue of 12. When we do this, we count to 12 five times, which equals 60.

As for the attri­bu­tion to Tes­la for the mul­ti­pli­ca­tion map, I think that Grether should have gone fur­ther back to 2000 BCE to those who first did math in base 60: the Sume­ri­ans. Base 60 has been around for over four thou­sand years. Even in the fourth cen­tu­ry BCE, the oth­er cul­tures world­wide, includ­ing the Ekari peo­ple of Indone­sia, used base 60. How­ev­er, base 60 has sig­nif­i­cant val­ue in our cur­rent math­e­mat­ics. It is the heart and soul of math­e­mat­ics. Base 60 is still used today to mea­sure geo­graph­ic coor­di­nates, deter­mine the time on our clocks, and study angles in geom­e­try and trigonom­e­try. When we fall into a deep sleep, our hearts beat at about 60 beats per minute. And, in a study done at Ruhr Uni­ver­si­ty Bochum, car­di­ol­o­gists found that music that plays at 60 beats per minute, like Mozart’s Sym­pho­ny No 40 in g minor, reduces stress and increas­es your relax­ation, which helps you to study and retain infor­ma­tion.[v]

So, I have decid­ed to re-attribute the name of Tesla’s Mul­ti­pli­ca­tion Map to the Sumer­ian Sex­a­ges­i­mal Spi­ral. It is offi­cial because the evi­dence of its use of base 60 is irrefutable. And it is offi­cial because I am the last per­son to name it so. So, thank you Dr. Stigler for the idea! This is my eponymic con­tri­bu­tion to the world of math, sci­ence, and history!

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[i] “The Mys­tery of The Tes­la Hoax,” Con­quer Maths, last mod­i­fied Jan­u­ary 18, 2017, https://www.conquermaths.com/news/post/index/395/The-Mystery-of-The-Tesla-Hoax.

[ii] Par­manand Singh, The so-called fibonac­ci num­bers in ancient and medieval India, His­to­ria Math­e­mat­i­ca, Vol­ume 12, Issue 3, 1985, Pages 229–244

[iii] Dhanan­jay Gopal et al., An Intro­duc­tion to Met­ric Spaces (Boca Raton: CRC Press, 2020), 51.

[iv] Stanko Bilin­s­ki, Über die Rhombenisoed­er, Peri­od­icum Math­e­mati­co-Physicum et Astro­nom­icum 15 (n.d.), 251–263.

[v] Hans-Joachim Trappe and Gabriele Voit, The Car­dio­vas­cu­lar Effect of Musi­cal Gen­res: A Ran­dom­ized Con­trolled Study on the Effect of Com­po­si­tions by W. A. Mozart, J. Strauss, and ABBA, PubMed Cen­tral (PMC), last mod­i­fied May 20, 2016.