Why Did Thomas Harriot Invent Binary?

From the early eighteenth century onward, primacy for the invention of binary numeration and arithmetic was virtually universally credited to the German polymath Gottfried Wilhelm Leibniz (1646–1716) (see, for instance, [5, p. 335] and [10, p. 74]). Then, in 1922, Frank Vigor Morley (1899–1980) famous that an unpublished manuscript of the English mathematician, astronomer, and alchemist Thomas Harriot (1560–1621) contained the numbers 1 to eight in binary. Morley’s solely remark was that this foray into binary was “definitely previous to the same old dates given for binary numeration” [6, p. 65]. Nearly thirty years later, John William Shirley (1908–1988) printed reproductions of two of Harriot’s undated manuscript pages, which, he claimed, confirmed that Harriot had invented binary numeration “almost a century earlier than Leibniz’s time” [7, p. 452]. However whereas Shirley accurately asserted that Harriot had invented binary numeration, he made no try to elucidate how or when Harriot had finished so. Curiously, few since Shirley’s time have tried to reply these questions, regardless of their apparent significance. In spite of everything, Harriot was, so far as we all know, the primary to invent binary. Accordingly, answering the how and when questions on Harriot’s invention of binary is the goal of this brief paper.

The story begins with the weighing experiments Harriot performed intermittently between 1601 and 1605. A few of these have been merely experiments to find out the weights of various substances in a measuring glass, reminiscent of claret wine, seck (i.e., sack, a fortified wine), and canary wine (see [3, Harriot, Add. Mss. 6788, 176r]), whereas different experiments have been supposed to find out the precise gravity, that’s, the relative density, of a wide range of substances.^{Footnote 1}

Listed below are three outcomes from Harriot’s experiments [3, Harriot, Add. Mss. 6788, 176r]:

Harriot’s technique of recording his measurements is the important thing to his invention of binary and so deserves some remark. Utilizing the troy system of measurement, he recorded the burden of every substance by decomposing it into ounces (typically utilizing the outdated image for ounces,
, a variant of the extra widespread ℥), then (frac{1}{2}) ounce, (frac{1}{4}) ounce, (frac{1}{8}) ounce, (frac{1}{16}) ounce, and at last grains. Since a troy ounce consists of 480 grains, the varied weights of his scale have the next grain values:

(frac{1}{2}) oz = 240 grains

(frac{1}{4}) oz = 120 grains

(frac{1}{8}) oz = 60 grains

(frac{1}{16}) oz = 30 grains

Collectively, the 4 part-ounce weights are 30 grains shy of 1 ounce, and certainly, in all of Harriot’s experiments, the measurement of grains by no means goes above 30. With this in thoughts, allow us to look once more at his report of weighing claret wine:

The primary quantity (14) is ounces, the ultimate quantity (24) grains, and the numbers in between check with half ounces—the (frac{1}{2}) within the (frac{1}{2}) ounce place indicating that the (frac{1}{2}) ounce weight was used, the 0 within the (frac{1}{4}) ounce place indicating that the (frac{1}{4}) ounce weight was not used, and many others.^{Footnote 2}

With regard to Harriot’s invention of binary, of explicit curiosity is one manuscript (reproduced beneath) that accommodates a report of a weighing experiment on the prime, and examples of binary notation and arithmetic on the backside. Listed below are the calculations from the weighing experiment, which was involved with discovering the distinction in capability between two measuring glasses [3, Harriot, Add. Mss. 6788, 244v]:

Observe right here that “troz” stands for “troy ounce.” Beneath all this, Harriot sketched a desk of the decimal numbers 1 to 16 in binary notation and labored out three examples of multiplication in binary: 109 × 109 = 11881, 13 × 13 = 169, and 13 × 3 = 39; see Determine 1.

Thomas Harriot’s binary multiplication [3, Harriot, Add. Mss. 6788, 244v]. Courtesy of the British Library Board.

As far as I do know, the one one who has tried to elucidate Harriot’s transition from weighing experiments to the invention of binary is Donald E. Knuth, who writes:

Clearly he [Harriot] was utilizing a steadiness scale with half-pound, quarter-pound, and many others., weights; such a subtraction was undoubtedly a pure factor to do. Now comes the flash of perception: he realized that he was basically doing a calculation with radix 2, and he abstracted the scenario [4, p. 241].

Whereas Knuth is mistaken in regards to the dimension of weights used, apparently lacking the abbreviation “troz” (= troy ounce) and taking the glyph
to check with pound relatively than ounce, his suggestion relating to Harriot’s “flash of perception” appears believable. However it’s potential to go additional, as a result of it’s unlikely that Harriot stumble on binary notation just because he was utilizing weights in a power-of-2 ratio, one thing that was a well-established apply on the time. Equally if no more necessary was the truth that he recorded the measurements made with these weights in a power-of-2 ratio too. For when recording the weights of the varied part-ounce measures, Harriot used a rudimentary type of positional notation, by which for each place he put down both the total place worth or 0, relying on whether or not or not the burden in query had been used. Therefore when weighing the primary “glass and water,” Harriot’s result’s equal to:

Or certainly, if we simply concentrate on the part-ounces and specific them as powers of two:

From such a way of recording weights in a power-of-2 ratio, it’s however a really small step to binary notation, by which, as an alternative of noting in every place both 0 or the total place worth, one merely places down both 0 or 1 relying on whether or not or not the burden in query was wanted. Harriot’s invention of binary due to this fact owed at the least as a lot to his personal idiosyncratic type of positional notation for recording part-ounce weights because it did to his use of these weights.

One oddity with Harriot’s “flash of perception” is that it didn’t lead him to binary expansions of reciprocals, which is what his notation is closest to. That’s, he didn’t symbolize (frac{1}{2}) ounce as [0].1, (frac{1}{4}) ounce as [0].01, (frac{1}{8}) ounce as [0.]001, or (frac{1}{16}) ounce as [0].0001. As a substitute, he continued to make use of decimal fractions to report the part-ounce weights in his weighing experiments. So though binary was an outgrowth of Harriot’s idiosyncratic technique of recording part-ounce weights, at no level did he use binary to report these weights. From that we could surmise that he didn’t suppose binary notation provided higher comfort or readability than his personal technique of recording part-ounce weights.

But Harriot was sufficiently intrigued by his new quantity system to discover it over an additional 4 manuscript pages, understanding how you can do three of the 4 fundamental arithmetic operations (all however division) in binary notation. On one sheet, Harriot wrote examples of binary addition (equal to 59 + 119 = 178 and 55 + 114 = 169) and subtraction (equal to 178 – 59 = 119 and 169 – 55 = 114) and the identical instance of multiplication in binary (109 × 109) as above, this time solved in two alternative ways (Harriot, Add. Mss. 6786, 347r). On a unique sheet, he transformed 1101101₂ to 109, calling the method “discount,” after which labored by way of the reciprocal course of, known as “conversion,” of 109 to 1101101₂ (Harriot, Add. Mss. 6786, 346v). On yet one more sheet, he jotted down a desk of 0 to 16 in binary, a easy binary sum: 100000 + [0]1[00]1[0] = 110010 (i.e., 32 + 19 = 51), and one other instance of multiplication, 101 × 111 = 100011 (i.e., 5 × 7 = 35) (Harriot, Add. Mss. 6782, 247r). And on a unique sheet once more (reproduced beneath), he drew a desk of 0 to 16 in binary, one other with the binary equivalents of 1, 2, 4, 8, 16, 32, and 64, gave a number of examples of multiplication in binary (equal to three × 3 = 9; 7 × 7 = 49; and 45 × 11 = 495), and produced a easy algebraic illustration of the primary few phrases of the powers of two geometric sequence (see Determine 2):

A web page of Thomas Harriot’s calculations. Within the backside left-hand nook will be seen the calculation of first few phrases of the powers of two geometric sequence reproduced within the textual content [3, Harriot, Add. Mss. 6786, 516v]. Courtesy of the British Library Board.

And on an additional sheet, Harriot employed a type of binary reckoning utilizing repeated squaring, combining this with floating-point interval arithmetic, so as to calculate the higher and decrease bounds of two²⁸²⁶² [3, Harriot, Add. Mss. 6786, 243v]; for additional particulars see [4, pp. 242–243]). The entire of Harriot’s work on binary is captured on the handful of manuscript pages described on this paper.

Now that we all know how Harriot arrived at binary, it stays to ask when he did so. Though Harriot typically recorded the date on his manuscripts, sadly he didn’t accomplish that on any of the manuscript pages that includes binary numeration. As such, it isn’t potential to find out the precise date of his invention, although it may be narrowed down, as we will see. Knuth conjectured that “Harriot invented binary arithmetic someday in 1604 or 1605” on the grounds that the manuscript containing a weighing experiment along with binary numeration and arithmetic is catalogued between one dated June 1605 and one other dated July 1604 [4, p. 241].

But as Knuth concedes, Harriot’s manuscripts will not be so as (as ought to be clear sufficient from the truth that one dated July 1604 follows one dated June 1605), so affixing a date to at least one manuscript primarily based on its place within the catalogue is problematic. As famous on the outset, Harriot’s weighing experiments started in 1601, certainly on September 22, 1601, and already in manuscripts from that yr he was utilizing his idiosyncratic technique of recording part-ounce weights (see [3, Harriot, Add. Mss. 6788 172r] and [176r]) that led to his pondering of binary, so it can’t be dominated out that binary was invented as early as September 1601. The most recent date for Harriot’s invention of binary might be November 1605, at which period Harriot’s patron, Henry Percy, ninth Earl of Northumberland (1564–1632), was imprisoned in reference to the Gunpowder Plot.

Round this time, Harriot, too, fell underneath suspicion of being concerned within the plot and was imprisoned for various weeks earlier than efficiently pleading for his freedom. After his launch, he didn’t resume his weighing experiments or, we could suppose, the investigations into binary that arose from them. That is maybe unsurprising. Whereas Leibniz noticed a sensible benefit in utilizing binary notation as an instance issues and theorems involving the powers of two geometric sequence (see [8]), Harriot seems to have handled binary as little greater than a curiosity with no sensible worth.

However, Harriot’s invention of binary is a startling achievement whenever you understand that the concept of exploring nondecimal quantity bases, versus tallying techniques, was not commonplace within the seventeenth century. Whereas counting in fives, twelves, or twenties was nicely understood and broadly practiced, the concept of numbering in bases aside from 10 was not. The trendy concept of a base for a positional numbering system was nonetheless coalescing, but it surely was conceived by just a few, with Harriot maybe the primary. Sadly, regardless of his nice perception, Harriot didn’t publish any of his work on binary, and his manuscripts remained unpublished till fairly lately, being scanned and put on-line in 2012–2015. Though Harriot rightly deserves the accolade of inventing binary many a long time earlier than Leibniz, his work on it remained unknown till 1922, and so didn’t affect Leibniz or anybody else, nor did it play any half within the adoption of binary as pc arithmetic within the Thirties (see [9]). That’s one accolade that also belongs to Leibniz.