American Presidents and their Math

Many of our presidents were trained in or used math at some point in their careers. An interesting note is that when someone has a productive disposition toward math, i.e. sees the value of and confidence in using mathematics to resolve problems they will use it to resolve many problems, not apparently related to math. It becomes a way of thinking. In this short note we look at some of the US presidents so disposed. Mathematical training was of course an important part of the curriculum as taught to many of our earlier presidents. They were schooled in algebra and geometry. Calculus is another matter. Let's look at some of them.

George Washington

• George Washington (1732-1799) was early in his career a surveyor. The mathematics of surveying includes foremost the techniques of planar measurement. These include the right triangle, oblique angles and triangles, azimuth, angles, bearing, bearing intersections, distance intersections, coordinate geometry, law of sines and cosines, interpolation, compass rules, horizontal and vertical curves, grades, and slopes. This sounds much like what is now the standard curriculum in four years of high school math.  Naturally, much of this involves algebra, the concept of a unknown, and the solving of algebraic equations.
  
  
  
  Thomas Jefferson
  
  
• Thomas Jefferson (1743-1826) invented a crypto system in 1795, just a few years before becoming president. It was so effective and secure (for the time) that it was used by the US Army from 1923-1942. The idea is remarkably simple. Take any number of disks with the letters of the alphabet on the outer rim, all mounted on an axle to make them rotatable. The disks were numbered, say from 1 to 10. The disks could be arranged in any order by restacking them. If the sender and receiver both know the correct stacking order a message could be transmitted. Here's how. Using a particular stacking order, the message is encrypted by rotating the disks to create a message. Now copy any row from the disks, looking like complete gibberish. Send this message. The receiver, stacks the disks properly and then rotates them to form the "gibberish" message, and then looks for the row that makes sense. The odds of there being two intelligible messages is remote. This can be checked first by the sender - just to be sure. The Jefferson disk had 36 discs. With just ten discs, this cypher is child's play to decrypt with modern computers, being there are only 10!= 3,628,800 possible rearragements of the disks. However, Jefferson method employed 36 disks. This implies there are 36!= 371993 326789901 217467999 448150835 200000000 ≐3. 7199×10⁴¹  possible rearrangements, making cracking it more difficult - impossibly difficult by hand. However, there are many decrypting tricks that could be employed to render this type of cypher not as difficult as it may appear. This cypher was re-invented by Etienne Bazeries a century later. Etienne Bazeries considered it uncrackable. One vulnerability of the encryption is that both parties need to know the key (i.e. the correct ordering of the disks). This is an example of a rotor encryption system, not unlike but far simpler than the World War II German encryption system, Enigma.   For a picture, please see http://en.wikipedia.org/wiki/Jefferson_disk.  To confirm Jefferson's interest in mathematics we add the following.
  
  
  
  From the writings of Thomas Jefferson, the letter to Thomas Lomax Monticello on Mar. 12, 1799 contains the following passage:
  DEAR SIR, -- I have to acknolege the receipt of your favor of May 14 in which you mention that you have finished the 6. first books of Euclid, plane trigonometry, surveying & algebra and ask whether I think a further pursuit of that branch of science would be useful to you. There are some propositions in the latter books of Euclid, & some of Archimedes, which are useful, & I have no doubt you have been made acquainted with them. Trigonometry, so far as this, is most valuable ...
  
  As is apparent, Jefferson is writing to Mr. Monticello about what materials he should read and understand in his personal advancement in scholarship. Jefferson also advises Mr. Monticello about other scientific subjects worthy of study. In another letter*, dated July 5, 1814, Jefferson talked about the value of formal education in the classics.
  
  But why am I dosing you with these Ante-diluvian topics? Because I am glad to have some one to whom they are familiar, and who will not receive them as if dropped from the moon. Our post-revolutionary youth are born under happier stars than you and I were. They acquire all learning in their mothers' womb, and bring it into the world ready-made. The information of books is no longer necessary; and all knolege which is not innate, is in contempt, or neglect at least. Every folly must run it's round; and so, I suppose, must that of self-learning, and self sufficiency; of rejecting the knolege acquired in past ages, and starting on the new ground of intuition. When sobered by experience I hope our successors will turn their attention to the advantages of education. I mean of education on the broad scale, and not that of the petty academies, as they call themselves, which are starting up in every neighborhood, and where one or two men, possessing Latin, and sometimes Greek, a knolege of the globes, and the first six books of Euclid, imagine and communicate this as the sum of science. They commit their pupils to the theatre of the world with just taste enough of learning to be alienated from industrious pursuits, and not enough to do service in the ranks of science. *The unusual spellings above are from the period.
  
• Abraham Lincoln (1809- 1865) was also a surveyor in his youth. Remarkably, having little formal schooling, Lincoln learned the required mathematics required independently. Lincoln was blessed with a prodigious memory, and a life-long passion for literature including Shakespeare, Burns, and Byron.  He also studied the law independently, though he never really immersed himself its subtleties.   It was no accident of fate he attended the Ford Theater the night of his assassination. He loved the theater and went often.
  
  
  Abraham Lincoln
  There is a most recent update to Lincoln and his math.  It is a ciphering notebook of Lincoln made when he was just 17 years old and it involves the Rule of Three, which even the ancient Chinese consider important enough to author a text on.  Basically, the rule of three goes like this:  if a is to b as c is to d, and you know three of these quantities, what it the value of the fourth.  This is precisely the subject now involving algebra, in it multitude of forms. The materials are copyrighted, and so I will simply give a link.  See   http://www.cbsnews.com/8301-201_162-57588329/math-pages-suggest-lincoln-had-more-education/ The investigators suggest Lincoln may have had far more schooling than his biographers suggest, even up to two full years.
  
  
• James A. Garfield (1831-1881) produced in 1876 an independent trapezoidal proof of the Pythagorean theorem: For any right triangle with legs a and  b, the hypotenuse is given by the formula c²=a²+b².  BTW, there are about 300 proofs if this, arguably the most famous theorem in mathematics. It was published in the New England Journal of Education. In his early years, he was a school teacher. He apparently enjoyed any tasks of a mathematical nature, such as when he was appointed chairman of a Subcommittee on the Census.  As a note, one does not just simply set at the desk one day and develop an original of anything.  Rest assured, Garfield had thought long on the matter and was well versed in the many proofs known at this time.
  
  
  James A. Garfield.
   

Other presidents that surely were well schooled in mathematics include courses in calculus.

• Herbert Hoover (1874-1964) was a Mining Engineer. 
• Jimmy Carter (1924-) was a Nuclear Engineer. 

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P.S.  If you want to get to really serious math, geometry in particular, consider Napoleon Bonaparte (1769-1821).  He invented original theorems in geometry quite far beyond the Pythagorean theorem to the extent that two of the absolute best French mathematicians of his day, Pierre Simon Laplace (1749-1827) and Joseph-Louis Lagrange(1736-1813), were duly impressed. They involved the so-called Mascheroni constructions. Laplace and Lagrange are even today revered as outstanding mathematicians in mathematics history.   It is difficult to imagine any contemporary politician either interested or capable in mathematical argument - regardless of the country.