Thomas Harriot and the New World


The moon was but a month old when Adam was no more,

and raught not to five weeks when he came to five score.



Nothing is known of Thomas Harriot (sometimes spelled Hariot) until he entered St. Mary’s College in Oxford university in 1577 as a commoner at the age of 17.  A friend later wrote “Tom Harriot had a far greater gift for language than I had. He enjoyed reading the writings of the ancient Romans, sharpening his language abilities through disputation and debate, and writing poetry in Latin." Harriot received a bachelor’s degree in 1580 – one of only three in his class to do so.  On the recommendation of the principal of St. Mary’s, Walter Raleigh hired Harriot to teach improved navigation techniques to Raleigh’s ship captains, to facilitate Raleigh’s plans for colonization in North America. Still only 22, Harriot had first to learn something about navigation himself, but he was a quick study, and soon compiled his knowledge in a private book called Arcticon.


Raleigh’s expedition to North America in 1584 had returned with two native Americans, Manteo and Wanchese, from whom Harriot learned the Algonquian language. Harriot even created a phonetic alphabet in order to transcribe the language. The following year, Harriot was sent on the next expedition to North America, which settled on Roanoke Island off the coast of what is now called North Carolina. (The English called it Virginia at the time.) One of their ships foundered off the coast, and the attempt to establish a permanent colony did not go well. A year later (1586) the men were evacuated back to England in ships of Sir Francis Drake. Many members of the expedition gave unfavorable accounts of the conditions at Roanoke. These reports threatened to undermine royal support for Raleigh’s plans, so Raleigh asked Harriot to write a more favorable account of the expedition. The result was Harriot’s “A Brief and True Report of the New Found Land of Virginia”. This was the only writing of Harriot’s that was formally published in his lifetime.


The Report consisted mostly of a catalogue of the land and resources in the region, but the final section is devoted to “the nature and manners of the people”. This gives a fascinating account of how the natives were bewildered and awestruck by the English:


Most things they saw with us, as mathematical instruments, sea compasses, the virtue of the loadstone in drawing iron, a perspective glass whereby was shown many strange sights, burning glasses, wildfire works, guns, books, writing and reading, spring clocks that seem to go of themselves, and many other things that we had, were so strange unto them, and so far exceeded their capacities to comprehend the reason and means how they should be made and done, that they thought they were rather the works of gods than of men... which made many of them to have such opinion of us, as that if they knew not the truth of god and religion already, it was rather to be had from us, whom God so specially loved, than from a people that were so simple, as they found themselves to be in comparison with us.


Harriot also briefly described the society of the natives and their living arrangements, making an interesting comment on their languages.


Their town are but small, some containing but 10 or 12 houses, some 20; the greatest we have seen had but 30 houses... In some places only one town belongeth to the government of a chief, in others some two or three. The greatest chief that yet we had dealing with had 18 towns in his government... The language of every government is different from any other, and the farther they are distant the greater is the difference...


It’s remarkable that, if Harriot’s observation is correct, the languages of the small native tribes were so fragmented, with each small cluster of people having a distinct language. Harriot went on to discuss the religious beliefs of the natives, and then makes a somewhat heart-breaking remark:


One other rare and strange accident, leaving others, will I mention before I end, which moved the whole country that either knew or heard of us, to have us in wonderful admiration.  There was no town where we had any subtle devise practiced against us but that within a few days after our departure from every such town the people began to die very fast, and many in a short space; in some towns about twenty, in some forty, in some sixty, and in one six score, which in truth was very many in respect of their numbers...  The disease also so strange, that they neither knew what it was, nor how to cure it; the like by report of the oldest men in the country never happened before, time out of mind.


These events caused some of the natives to try to curry favor with the English, while others turned against them. For example, after arriving back in Roanoke, Manteo continued to co-operate with the English, but Wanchese left their company and preached opposition. The natives struggled to comprehend what was happening.


In all the space of their sickness, there was no man of ours known to die, or that was specially sick; they noted also that we had no women amongst us, neither that we did care for any of theirs.  Some therefore were of the opinion that we were not borne of women, and therefore not mortal, but that we were men of an old generation many years past, then risen again to immortality.  Some would seem to prophesy that there were more of our generation yet to come, to kill theirs and take their places... those that were immediately to come after us they imagined to be in the air, yet invisible and without bodies, and that by our entreaty and for the love of us did make the people to die in that sort as they did by shooting invisible bullets into them.


After staying a year on Roanoke (with a side expedition to Chesapeake Bay), amid reports that the Spanish might be preparing to attack the settlement, and unhappy with their living conditions, the English decided to return home on Sir Francis Drake’s fleet which had arrived from the Caribbean. The next year, Raleigh sent another party, this time including women, to found a permanent settlement on Roanoke. After delivering the settlers, the ships departed, promising to return in six months with supplies, but due to various circumstances (including the attack of the Spanish Armada), no supply ship was able to return to Roanoke until three years had passed. By that time, all the settlers had vanished. The fate of the “lost colony” remains a mystery, although there was some evidence that a few of the settlers may have been absorbed into the native tribes.


After arriving back in England in 1586, Harriot worked on his Brief and True Report, published in 1588, and also participated in Raleigh’s colonization project in Ireland. In 1592, Raleigh fell out of favor with Queen Elizabeth (partly due to his unauthorized marriage to Bess Throckmorton) and was temporarily imprisoned in the Tower of London. Although remaining close to Raleigh, Harriot acquired another patron, Henry Percy (9th Earl of Northumberland), around 1595, and devoted himself to mathematical and scientific studies.


From his study of celestial observations, both for navigational and for scientific purposes, Harriot was interested in spherical trigonometry, and he was the first to discover and prove a beautiful formula (now often called Girard’s formula) for the area of a spherical triangle in terms of the surface angles at the vertices. A spherical triangle is a region on the surface of a sphere bounded by three great circle segments (i.e., geodesics). For a sphere of radius R, and an arbitrary spherical triangles with angles α,β,γ, Harriot’s formula is



which is easily seen by considering the figure below.



The entire surface area of the sphere is S = 4πR2. The area of the yellow and gray regions is (2α/2π)S = 4αR2, the area of the green and gray regions is (2β/2π)S = 4βR2, and the area of the blue and gray regions is (2γ/2π)S = 4γR2. Adding these three regions gives the total area of the sphere, but with four extra gray regions, because they are included in each of the three regions. Therefore, letting A denote the area of one gray region, we have (4α + 4β + 4γ)R2 = (4π)R2 + 4A, and hence A = (α + β + γ − π)R2.


This formula gives the curvature of the surface entirely in terms of the surface area and surface angles of the triangles, without any measurements outside the surface, so it could be regarded as a precursor of Gauss’s concept of intrinsic curvature and non-Euclidean geometry. Harriot made some of the first maps of regions he visited in North America, dividing the territory into triangular regions, and he would certainly have been conscious of the fact that, for large enough triangles (in comparison with the radius of the Earth) the sum of the angles would exceed π. This is reminiscent of Gauss’s surveying in Germany over two centuries later, measuring the angles of the triangle formed by three prominent mountain peaks.


Harriot made important discoveries in algebra. Perhaps the most significant was his original treatment of polynomials as the products of linear factors. He recognized the usefulness, not only of writing equations in symbolic form, but in bringing all the terms over to one side, and equating that expression to zero. This was crucial for the next step, which was to factor the expression. For example, he explained that a cubic polynomial such as x3 + Ax2 + Bx + C = 0 has solutions a,b,c, then the polynomial could be written as (x−a)(x−b)(x−c) = 0, and therefore A = −(a+b+c), B = (ab+bc+ca), and C = −abc.


He was also the first to derive fully general formulas for the sum of the nth powers of the first N integers, and this led him to useful interpolation formulas, implicitly leading to the Bernoulli numbers. His manuscripts also show that he considered the binary number system, investigated many combinatoric problems and what is now called Pascal’s triangle, developed analytic geometry in rectangular coordinates, recognized the correspondence between geometrical curves and algebraic expressions (prior to Descartes), dealt with imaginary and complex roots of equations, and studied the sphere-packing problem and what later was called Kepler’s conjecture on the densest packing of spheres.


Harriot was prompted to study the sphere-packing problem by Raleigh, who asked him to calculate how many cannonballs could be packed into a given volume, but Harriot viewed the question on a deeper level, in the context of his atomistic view of physics. The revival of atomism in modern Europe is often attributed to Gassendi in the 1630’s, but it’s clear that Harriot was already a vocal proponent of atomism decades earlier. He had also made a study of optics, and discovered what is now called Snell’s law of refraction. He corresponded with Johannes Kepler on both of these subjects, optics and sphere packing, and provided him with tables of optical data. Harriot also tried to convince Kepler that the transparency of dense substances such as glass and water implies that they are mostly empty:


A dense disphanous body, therefore, which to the sense appears to be continuous in all parts, is not actually continuous. But it has corporeal parts which resist the rays, and incorporeal parts vacua which the rays penetrate. So that refraction is nothing else than an internal reflection, and the part of the rays which are received inside, although to the sense it appears straight, is nevertheless composed of many straight line segments.


Kepler had written to Harriot because the only works on optics Kepler could find were unreliable. Kepler wrote “I was amazed by many things in the tables [of refraction], which I had been ignorant of before... I think you have extended to me a key for seeking into the hidden natures of bodies.” However, Kepler was not persuaded by Harriot’s atomism. Instead, Kepler argued that transparency could be an inherent quality of some (continuous) substances. Harriot replied (in a letter of July 13, 1608) “If those assumptions and reasons satisfy you, I am amazed. I do not understand transparency unless caused by vacancy...” He went on to point out that “gold is among our elements in composition held to be most dense and opaque”, and yet if it is pounded to a sufficiently thin foil we can see the light of a candle through it, although the light is green in color. More than this, Harriot did not feel at liberty to say, since his scientific ideas left him open to charges of atheism, for which he and both his patrons had been investigated. Harriot was suspected of having argued that there were men before Adam.


Both Raleigh and Percy were imprisoned in the Tower, and Harriot himself had been briefly imprisoned, partly due to political intrigues after the death of Elizabeth and ascension of James I, and partly due to the fallout from the gunpowder plot. Harriot complained to Kepler


For things are in such a pass with us, that still yet I may not freely philosophize. Still yet we stick in the mire. I hope the Good God will make an end to these things shortly. After which better things are to be expected...


Harriot also made pioneering astronomical observations, having acquired a telescope at about the same time as Galileo. Among Harriot’s papers have been found the earliest known maps of the Moon. He was also the first to observe Sun spots, and used the observations to determine the rotation rate of the Sun. He may have been the first to determine the periods of the orbits of Jupiter’s moons, and made careful observations of the great comet of 1607, which later came to be called Halley’s comet. Some have suggested that it ought to have been named Harriot’s comet, since Harriot apparently inferred from his observations that the comet had an elliptical orbit, consistent with Kepler’s law for the planets.


Harriot never published any of his discoveries, for reasons which are not known. It certainly wasn’t for lack of encouragement, since his friends urged him to publish. This is from a letter from Sir William Lower to Harriot in 1610:


Do you not here startle, to see every day some of your inventions taken from you?  For I remember long since you told me as much, that the motions of the planets were not perfect circles. So you taught me the curious way to observe weight in Water, and within a while after Ghetaldi comes out with it in print, a little before Vieta prevented you of the garland of the great Invention of Algebra.  All these were your due, and many others that I could mention, and yet too great reservedness had robbed you of these glories.  But although the inventions be great, the first and last I mean, yet when I survey your storehouse, I see they are the smallest things, and such as in comparison of many others are of small or no value. Only let this remember you, that it is possible by too much procrastination to be prevented in the honor of some of your rarest inventions and speculations. Let your Country and friends enjoy the comforts they would have in the true and great honor you would purchase your self by publishing some of your choice works, but you know best what you have to do. Only I, because I wish you all good, with this, and sometimes the more longingly, because in one of your letters you gave me some kind of hope thereof.


Whatever the reason, Harriot’s failure to publish resulted in him being largely forgotten following his death. An abbreviated collection of his writings on algebra was published posthumously by his friends, in a book entitled Artis Analyticae Praxis, which was influential for some years, but it omitted many of the most advanced and original features of his work, which remained buried in his papers. For many years his papers were assumed to have been lost, but in 1784 a descendant of Henry Percy found thousands of pages of notes in what had been Percy’s country estate. These writings, however, have never yet been edited or published – which would be a difficult task, considering their fragmentary and disorganized form. But enough has been gleaned by scholars examining the pages to prompt a reappraisal of Harriot’s importance, and to at least add his name in a footnote when mentioning the discoveries that have been attributed to later scientists.


It’s interesting that none of the English men in Harriot’s party in Roanoke were “specially sick”, and yet they apparently transmitted some diseases to the natives. There is speculation that the English might have carried low grade measles, small pox, or influenza, for which the natives had no resistance. This is plausible, although we might wonder why the natives didn’t transmit some native diseases to the English in turn. Perhaps the lower population density in North America led to fewer and less virulent communicable diseases, although there is also evidence that the early explorers did indeed bring back some new diseases to Europe from the Americas.


Even aside from diseases, the English explorers definitely brought back tobacco. In his Brief and True Report Harriot wrote


There is an herb which is sowed apart by itself and is called by the inhabitants Vppówoc. The Spaniards generally call it Tobacco. The leaves thereof being dried and brought into powder, they use to take the fume or smoke thereof by sucking it through pipes made of clay into their stomach and head, from whence it purgeth superfluous fleame and other gross humors, openeth all the pores and passages of the body. The use thereof not only preserveth the body from obstructions but also... their bodies are notably preserved in health, and know not many grievous diseases wherewithal we in England are oftentimes afflicted. This Vppówoc is of so precious estimation amongest the natives, that they think their gods are marvelously delighted therewith, whereupon sometime they make hallowed fires and cast some of the powder therein for a sacrifice... We ourselves during the time we were there used to suck it after their manner, as also since our return, and have found many rare and wonderful experiments of the virtues thereof; of which the relation would require a volume by itself. The use of it by so many of late, men and women of great calling as else, and some learned Physicians also, is sufficient witness.


In 1613 Harriot developed cancer of the lips and nose. It gradually worsened over the years, and finally killed him in 1621. (Invisible bullets.)


It’s interesting that the years between when Harriot returned from North America and when he became afflicted with cancer, i.e., between about 1589 and 1613, are precisely the years during which the plays of William Shakespeare were written. Harriot was a friend of another playwright, Christopher Marlowe, and they resided in the same city, so it’s quite possible that Shakespeare and Harriot might have met. Although Harriot was an Oxford graduate, intimately familiar with court intrigue, and an exact contemporary of Shakespeare, with a known aversion to publication, no one (as far as I know) has ever suggested Harriot as the real author of Shakespeare’s plays. This is odd, considering that almost everyone else in Elizabethan England has been proposed as the author of those plays, and considering that Harriot’s chronology actually fits better than any of the other candidates. On the other hand, Harriot’s writing in the Brief and True Report shows no resemblance to Shakespeare, and the 5000 manuscript pages of scientific and mathematical notes left by Harriot seem to give solid evidence that he was otherwise occupied during those years.


Some have argued that Shakespeare’s early play, Love’s Labour's Lost, contained a satire of Raleigh’s circle of intellectual friends, and that the comically pedantic character of Holofernes represents Harriot. However, other accounts of Harriot’s character (that he was a warm, friendly, and likeable man) don’t seem consistent with this theory. Incidentally, Shakespeare’s play also contains a famous line about the “school of night”, which some have thought may be an allusion to Raleigh’s circle, supposedly taught by Harriot. (“Oh paradox! Black is the badge of hell, the hue of dungeons, and the school of night...”) However, the word “school” is sometimes rendered as “suite” or “shade”, and little credence is given to this speculation.


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