Jefferson's enthusiasm for mathematics was manifest at a number of levels. Beyond various utilitarian motives for studying mathematics, from the fact that it is a good training for the law to its well-rehearsed uses in a colonial frontier society, we can see that the study of mathematics influenced Jefferson's very way of thinking, in the very forms of expression and categories his mind thought in. And following on from that is something which his public role made possible, his promotion of mathematics Monticello as a critical component of education in a free democratic society.

Let us start with an example of what one might call utilitarian mathematics, albeit with a Jeffersonian twist. First, see Jefferson watching the ground being cleared for his new house, Monticello, in 1768. Anyone else in that situation would be fussing round making tea, but Jefferson was doing an exercise in rule of three:

Julius Shard fills the two-wheeled barrow in 3 minutes and carries it 30 yards in 1 1/2 minutes more. Now this is four loads of the common barrow with one wheel. So suppose that the four loads put in, in the same time, viz., 3 minutes, 4 trips will take 1 1/2 minutes—6, which added to 3 filling is—9 to fill and carry the same earth which was filled and carried in the two-wheeled barrow in 4 1/2 minutes. From a trial I made with the same two-wheeled barrow, I found that a man would dig and carry to the distance of 50 yards, 5 cubical yards of earth in a day of 12 hours' length. Ford's Phil did it, not overlooked, and having to mount his loaded barrow up a bank 2 feet high and tolerably steep. [Garden Book, pp.33-34]

There is something timeless about this sort of thing--work supervisors in Uruk or Nippur, three thousand years before, were doing exactly the same calculation on exactly the same subject. Jefferson's enthusiasm for a constant arithmetical monitoring of what was going on around him was carried to remarkable lengths: even when a few years later his closest friend Dabney Carr died, and Jefferson was preparing what is now the graveyard at Monticello to receive his body, he subsumed his deep sorrow and grief in further Babylonian-style calculations.

2 hands grubbed the graveyard 80 feet square—1/7 of an acre in 3 1/2 hours, so that one would have done it in 7 hours, and would grub an acre in 49 hours—4 days. [Memorandum Book, 23 May 1773]

Of course, one hardly needs an expensive education in Newtonian mathematics at William and Mary College to do that. But another of Jefferson's ventures truly shows the benefits of a Newtonian education.

Travelling through France fifteen years later, in 1788, he noticed peasants near Nancy ploughing, and fell to wondering about the design of the moldboard, that is, the surface which turns the earth: he spent the next ten years working on this, on and off, wondering how to achieve the most efficient design, both offering least frictional resistance, and which also would be easy for farmers out in the frontiers to construct, far from technical help. He consulted the Pennsylvania mathematician Robert Patterson (born in Ireland in 1743), and consulted also another Philadelphia luminary, the self-taught astronomer and mathematical instrument-maker David Rittenhouse (1732-1796). It transpired that the answer lay in one of Jefferson's old college textbooks, Emerson's Doctrine of Fluxions, in material deriving from the discussion of 'solids of least resistance' in Newton's Principia, Book ii. Jefferson's account appeared in the 1799 volume of the Transactions of the American Philosophical Society, of which he was president by this time. This is quite a good example of Newtonian mathematics in action, its perhaps surprising applicability to frontier needs, and of Jefferson's command of it. The most important thing isn't so much his solving the problem as his coming to see that there was a connection between his mathematical studies at William and Mary College and the furthering of frontier agriculture: that mathematics was the kind of thing to bear on the problems of farmers in the new country.

Several other examples of Jefferson's mathematics may be mentioned more briefly.

His successful advocacy of a method for apportionment of representatives in Congress was a delicate exercise in justice and the mathematics of representation which won out over Hamilton's quota method only after Washington had vetoed the latter. [This is discussed in some detail in Cohen 1995, 88-97.]
Another interesting topic is his work as secretary of state in the 1790s towards establishing standards of weights and measures across the United States. His proposal here was for a standard of length based upon a pendulum, as a natural phenomenon that was reproducible everywhere. He was very critical of the new French metric system of measure which was based as he saw it on the contingencies of French geography, a standard accessible in principle only to the French and without the universal quality that he advocated.
In similar vein too, but more successfully, he worked on what the best currency would be for the new country, coming down on the side of a dollar with decimal subdivisions after inquiring deeply into the relevant factors of ease of calculation, ease of learning, ease of trade, and a range of historical precedents and considerations.
It was while he was in France, at the time of the French Revolution, that Jefferson was writing to Tom Paine about bridge-building, of all things, and argued for the optimally stable bridge to have arches in the form of a catenary: indeed, Jefferson is the first person we know of to use the word 'catenary' in English.

^{Jefferson's letter to Thomas Paine, December 23^rd 1788}

And yet another area of his concerns worth exploring is Jefferson's venture into cryptography, a need he experienced as Secretary of State, as a consequence of which he invented a wheel cipher which was to be re-invented by the United States military around the time of the first world war.

^{Jefferson's cipher}

Throughout his life, Jefferson showed an interest in astronomy and attempted to make observations. A letter to David Rittenhouse about the 1778 solar eclipse indicates the level of his engagement and his attempts to improve his instruments [TJ to David Rittenhouse, July 19, 1778].

We were much disappointed in Virginia generally on the day of the great eclipse, which proved to be cloudy. In Williamsburgh, where it was total, I understand only the beginning was seen. At this place which is in Lat. 38x-8' and Longitude West from Williamsburgh about 1x-45' as is conjectured, eleven digits only were supposed to be covered, as it was not seen at all till the moon had advanced nearly one third over the sun's disc. Afterwards it was seen at intervals through the whole. The egress particularly was visible. It proved however of little use to me for want of a time piece that could be depended on; which circumstance, together with the subsequent restoration of Philadelphia to you, has induced me to trouble you with this letter to remind you of your kind promise of making me an accurate clock; which being intended for astronomical purposes only, I would have divested of all apparatus for striking or for any other purpose, which by increasing it's complication might disturb it's accuracy. A companion to it, for keeping seconds, and which might be moved easily, would greatly add to it's value. The theodolite, for which I spoke to you also, I can now dispense with, having since purchased a most excellent one.

And in his later years, Jefferson revisited the practical mathematics of his youth, for example with dialling exercises, still keen to calculate the lines of a sun-dial [TJ to Charles Clay, August 23, 1811]

I have amused myself with calculating the hour lines of an horizontal dial for the latitude of this place, which I find to be 37^o 22' 26". The calculations are for every five minutes of time, and are always exact to within less than half a second of a degree.

But he was not blinded to the limitations of his studies of a subject he had so little time to cultivate in depth, writing to Robert Patterson a few months later [TJ to Robert Patterson, November 10, 1811]:

Before I entered on the business of the world I was much attached to Astronomy & had laid a sufficient foundation at College to have pursued it with satisfaction & advantage. But after 40. years of abstraction from it, and my mathematical acquirements coated over with rust, I find myself equal only to such simple operations & practices in it as serve to amuse me. But they give me great amusement, and the more as I have some excellent instruments.

So we can see a range of particular instances where his mathematical training and cast of mind was used, to a greater or lesser extent, throughout his life. And Jefferson's most celebrated text, written some thirty-five years before, also turns out to have been informed by his mathematical experiences.