https://fermatslibrary.com/s/why-did-thomas-harriot-invent-binary FERMAT'S LIBRARY * Journal Club * Librarian * Margins * Log in Join our newsletter to receive a new paper every week [ ] Close [Subscribe!] Comments Ask a question or post a comment about the paper Join the discussion! Ask questions and share your comments. Sign in with Google Sign in with Facebook Sign in with email Thomas Harriot was a 16th-century polymath who made significant con... #### TL;DR The binary system, which is a system of representing ... The troy system of measurement is a system of weights that is used ... You can learn more about Leibniz's work on binary in this extensive... You can find more pages of Harriot's manuscripts here: [Manuscripts... > ***"Although Harriot rightly deserves the accolade of inventing b... [wKIUO] Vol.:(0123456789) The Mathematical Intelligencer [?] (c) The Author(s) 2023 https://doi.org/10.1007/s00283-023-10271-9 1 Why Did Thomas Harriot Invent Binary? LloydStrickland F rom the early eighteenth century onward, pri- macy for the invention of binary numeration and arithmetic was almost universally credited to the German polymath Gottfried Wilhelm Leibniz (1646-1716) (see, for example, [5, p. 335] and [10, p. 74]). Then, in 1922, Frank Vigor Morley (1899-1980) noted that an unpublished manuscript of the English mathematician, astronomer, and alchemist Thomas Harriot (1560-1621) contained the numbers 1 to 8 in binary. Morley's only comment was that this foray into binary was "certainly prior to the usual dates given for binary numeration" [6, p. 65]. Almost thirty years later, John William Shirley (1908-1988) published reproduc- tions of two of Harriot's undated manuscript pages, which, he claimed, showed that Harriot had invented binary numeration "nearly a century before Leibniz's time" [7, p. 452]. But while Shirley correctly asserted that Harriot had invented binary numeration, he made no attempt to explain how or when Harriot had done so. Curiously, few since Shirley's time have attempted to answer these ques- tions, despite their obvious importance. After all, Harriot was, as far as we know, the rst to invent binary. Accord- ingly, answering the how and when questions about Har- riot's invention of binary is the aim of this short paper. The story begins with the weighing experiments Harriot conducted intermittently between 1601 and 1605. Some of these were simply experiments to determine the weights of dierent substances in a measuring glass, such as claret wine, seck (i.e., sack, a fortied wine), and canary wine (see [3, Harri ot, Add. Mss. 6788, 176r]), while other experi- ments were intended to determine the specic gravity, that is, the relative density, of a variety of substances. Here are three results from Harriot's experiments [3, Harri ot, Add. Mss. 6788, 176r]: Claret wine 14 1 2 0 1 8 0 24g Seck 14 1 2 0 1 8 1 16 6 gr. Canary wine 14 1 2 1 4 0 0 24 gr. Harriot's method of recording his measurements is the key to his invention of binary and so deserves some com- ment. Using the troy system of measurement, he recorded the weight of each substance by decomposing it into ounc- es (sometimes using the old symbol for ounces, , a variant of the more common ), then 1 2 ounce, 1 4 ounce, 1 8 ounce, 1 16 ounce, and nally grains. Since a troy ounce is composed of 480 grains, the various weights of his scale have the follow- ing grain values: 1oz = 480 grains 1 2 oz = 240 grains 1 4 oz = 120 grains 1 8 oz = 60 grains 1 16 oz = 30 grains Together, the four part-ounce weights are 30 grains shy of one ounce, and indeed, in all of Harriot's experiments, the measurement of grains never goes above 30. With this in mind, let us look again at his record of weighing claret wine: Claret wine 14 1 2 0 1 8 0 24g The rst number (14) is ounces, the nal number (24) grains, and the numbers in between refer to part ounces-- the 1 2 in the 1 2 ounce position indicating that the 1 2 ounce weight was used, the 0 in the 1 4 ounce position indicating that the 1 4 ounce weight was not used, etc. With regard to Harriot's invention of binary, of par- ticular interest is one manuscript (reproduced below) that contains a record of a weighing experiment at the top, and examples of binary notation and arithmetic at the bottom. Here are the calculations from the weighing experiment, which was concerned with nding the dierence in capac- ity between two measuring glasses [3, Harri ot, Add. Mss. 6788, 244v]: In the latter case, Harriot works out the relative density of materials such as brown mortar, copper ore, and lapis calaminaris (calamine) by the Archimedean method of weighing them rst in air and then in water, then working out the dierence between the two weights before dividing the weight in air by the dierence to determine the specic gravity (for more details on Harriot's experiments and specic gravity, see [2]). Clucas claims that Harriot's "weighing is done to the highest degree of accuracy in ounces, drachms, scruples and grains" [1, p. 124]. But this is clearly not the case. In the troy system, one ounce is equivalent to 8 drachms, and each drachm in turn equivalent to 3 scruples (with each scruple worth 20 grains). Yet Harriot's measurements divide the ounce into 16, not 8 (drachms) or 24 (scru- ples), indicating that the weights he was using were simply 1 2 ounce, 1 4 ounce, 1 8 ounce, etc. [iD7nBUtSHk] [?] The Mathematical Intelligencer 2 troz. A. Rounde measuring glasse weyeth dry 3 1 2 0 1 8 1 16 + 21 gr. B. The other rounde measure 3 0 1 4 1 8 1 16 +5 gr. A. Glasse & water 11 0 0 1 8 0 + 28 gr. 3 1 2 0 1 8 1 16 + 21 Water 7 0 1 4 1 8 1 16 + 7 gr. B. Glasse & water 10 1 2 0 0 1 16 + 10 gr. 3 0 1 4 1 8 1 16 + 5 Water 7 0 0 1 8 0 5 di. 1 4 0 1 16 + 2 gr. Note here that "troz" stands for "troy ounce." Underneath all this, Harriot sketched a table of the decimal numbers 1 to 16 in binary notation and worked out three examples of multiplication in binary: 109 x 109 = 11881, 13 x 13 = 169, and 13 x 3 = 39; see Figure1. So far as I know, the only person who has attempted to explain Harriot's transition from weighing experiments to the invention of binary is Donald E. Knuth, who writes: Clearly he [Harriot] was using a balance scale with half-pound, quarter-pound, etc., weights; such a sub- traction was undoubtedly a natural thing to do. Now comes the flash of insight: he realized that he was essentially doing a calculation with radix 2, and he abstracted the situation [4, p. 241]. While Knuth is mistaken about the size of weights used, apparently missing the abbreviation "troz" (= troy ounce) and taking the glyph to refer to pound rather than ounce, his suggestion regarding Harriot's "ash of insight" looks plausible. But it is possible to go further, because it is unlikely that Harriot hit upon binary nota- tion simply because he was using weights in a power-of-2 ratio, something that was a well-established practice at the time. Equally if not more important was the fact that he recorded the measurements made with these weights in a power-of-2 ratio too. For when recording the weights of the various part-ounce measures, Harriot used a rudimentary form of positional notation, in which for every position he put down either the full place value or 0, depending on whether or not the weight in question had been used. Hence when weighing the rst "glass and water," Harriot's result is equivalent to: Position: Ounces 1 2 ounces 1 4 ounces 1 8 ounces 1 16 ounces Grains Harriot's measure- ment: 11 0 0 1 8 0 28 Or indeed, if we just focus on the part-ounces and express them as powers of 2: 2 -1 ounce 2 -2 ounce 2 -3 ounce 2 -4 ounce 0 0 2 - 3 ounce 0 From such a method of recording weights in a power-of-2 ratio, it is but a very small step to binary notation, in which, instead of noting in each position either 0 or the full place value, one simply puts down either 0 or 1 depending on whether or not the weight in question was needed. Harriot's invention of binary therefore owed at least as much to his own idiosyncratic form of positional notation for recording part- ounce weights as it did to his use of those weights. One oddity with Harriot's "ash of insight" is that it did not lead him to binary expansions of reciprocals, which is what his notation is closest to. That is, he did not represent 1 2 ounce as [0].1, 1 4 ounce as [0].01, 1 8 ounce as [0.]001, or 1 16 ounce as [0].0001. Instead, he continued to use decimal fractions to record the part-ounce weights in his weigh- ing experiments. So although binary was an outgrowth of Harriot's idiosyncratic method of recording part-ounce weights, at no point did he use binary to record these weights. From that we may surmise that he did not think binary notation oered greater convenience or clarity than his own method of recording part-ounce weights. Yet Harriot was suciently intrigued by his new num- ber system to explore it over a further four manuscript pag- es, working out how to do three of the four basic arithmetic operations (all but division) in binary notation. On one sheet, Harriot wrote examples of binary addition (equiva- lent to 59 + 119 = 178 and 55 + 114 = 169) and subtrac- tion (equivalent to 178 - 59 = 119 and 169 - 55 = 114) and the same example of multiplication in binary (109 x 109) as above, this time solved in two dierent ways (Harri ot, Add. Mss. 6786, 347r). On a dierent sheet, he converted 1101101 2 to 109, calling the process "reduction," and then worked through the reciprocal process, called "conversion," of 109 to 1101101 2 (Harri ot, Add. Mss. 6786, 346v). On yet another sheet, he jotted down a table of 0 to 16 in binary, a simple binary sum: 100000 + [0]1[00]1[0] = 110010 (i.e., 32 + 19 = 51), and another example of multiplication, 101 x 111 = 100011 (i.e., 5 x 7 = 35) (Harri ot, Add. Mss. 6782, 247r). And on a dierent sheet again (reproduced below), he drew a table of 0 to 16 in binary, another with the bi- nary equivalents of 1, 2, 4, 8, 16, 32, and 64, gave several examples of multiplication in binary (equivalent to 3 x 3 = 9; 7 x 7 = 49; and 45 x 11 = 495), and produced a simple algebraic representation of the rst few terms of the powers of 2 geometric sequence (see Figure2): b. a. aa b aaa bb aaaa bbb 1. 2. 4. 8. 16. 1 2 2 [ x ] 2 1 2 [ x ] 2 [ x ] 2 1 [ x ] 1 2 [ x ] 2 [ x ] 2 [ x ] 2 1 [ x ] 1 [ x ] 1 And on a further sheet, Harriot employed a form of binary reckoning using repeated squaring, combining this with oating-point interval arithmetic, in order to calculate the upper and lower bounds of 2 28262 [3, Harri ot, Add. Mss. 6786, 243v]; for further details see [4, pp. 242-243]). The whole of Harriot's work on binary is captured on the hand- ful of manuscript pages described in this paper. [8HjlLr0ffS] The Mathematical Intelligencer [?] 3 Figure1 Thomas Harriot's binary multiplication [3, Harri ot, Add. Mss. 6788, 244v]. Courtesy of the British Library Board. [A6TbiFOh2C] [?] The Mathematical Intelligencer 4 Figure2 A page of Thomas Harriot's calculations. In the bottom left-hand corner can be seen the calculation of rst few terms of the powers of two geometric series reproduced in the text [3, Harri ot, Add. Mss. 6786, 516v]. Courtesy of the British Library Board. The Mathematical Intelligencer [?] 5 Now that we know how Harriot arrived at binary, it remains to ask when he did so. Although Harriot often recorded the date on his manuscripts, unfortunately he did not do so on any of the manuscript pages featuring binary numeration. As such, it is not possible to determine the exact date of his invention, though it can be narrowed down, as we shall see. Knuth conjectured that "Harriot invented binary arithmetic one day in 1604 or 1605" on the grounds that the manuscript containing a weighing experiment together with binary numeration and arithme- tic is catalogued between one dated June 1605 and another dated July 1604 [4, p. 241]. Yet as Knuth concedes, Harriot's manuscripts are not in order (as should be clear enough from the fact that one dated July 1604 follows one dated June 1605), so affixing a date to one manuscript based on its position in the catalogue is problematic. As noted at the outset, Harriot's weighing experiments began in 1601, indeed on September 22, 1601, and already in manuscripts from that year he was using his idiosyncratic method of recording part-ounce weights (see [3, Harri ot, Add. Mss. 6788 172r] and [176r]) that led to his thinking of binary, so it cannot be ruled out that binary was invented as early as September 1601. The latest date for Harriot's in- vention of binary is probably November 1605, at which time Harriot's patron, Henry Percy, 9th Earl of Northum- berland (1564-1632), was imprisoned in connection with the Gunpowder Plot. Around this time, Harriot, too, fell under suspicion of being involved in the plot and was imprisoned for a number of weeks before successfully pleading for his freedom. After his release, he did not resume his weighing experiments or, we may suppose, the investigations into binary that arose from them. This is perhaps unsurprising. Whereas Leibniz saw a practical advantage in using binary notation to illustrate problems and theorems involving the powers of 2 geometric sequence (see [8]), Harriot appears to have treated binary as little more than a curiosity with no practical value. Nevertheless, Harriot's invention of binary is a startling achievement when you realize that the idea of exploring nondecimal number bases, as opposed to tallying systems, was not commonplace in the seventeenth century. While counting in ves, twelves, or twenties was well understood and widely practiced, the idea of numbering in bases other than 10 was not. The modern idea of a base for a posi- tional numbering system was still coalescing, but it was conceived by a few, with Harriot perhaps the rst. Unfor- tunately, despite his great insight, Harriot did not publish any of his work on binary, and his manuscripts remained unpublished until quite recently, being scanned and put online in 2012-2015. Although Harriot rightly deserves the accolade of inventing binary many decades before Leibniz, his work on it remained unknown until 1922, and so did not inuence Leibniz or anyone else, nor did it play any part in the adoption of binary as computer arithmetic in the 1930s (see [9]). That is one accolade that still belongs to Leibniz. Acknowledgments I would like to thank Owain Daniel Jones, Donald E. Knuth, Harry Lewis, and two anonymous referees for their helpful comments on an earlier version of this article. I would also like to thank the Gerda Henkel Stiftung, Dusseldorf, for the award of a research scholarship (AZ 46/V/21), which made this article possible. Open Access This article is licensed under a Creative Com- mons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropri- ate credit to the original author(s) and the source, pro- vide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Crea- tive Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References [1] Stephen Clucas. Thomas Harriot and the eld of knowl- edge in the English Renaissance. In Thomas Harriot: An Elizabethan Man of Science, edited by Robert Fox, pp. 93-136. Ashgate, 2000. [2] Stephen Clucas. "The curious ways to observe weight in Water": Thomas Harriot and his experiments on specic gravity. Early Science and Medicine 25:4 (2020), 302-327. [3] Thomas Harriot. Digital edition of manuscripts held by the British Library and Petworth House, edited by Jac- queline Stedall, Matthias Schemmel, and Robert Goulding. Available at ECHO (European Cultural Heritage Online): https:// echo. mpiwg- berlin. mpg. de/ conte nt/ scien tic_ revol ution/ harri ot/ harri ot_ manus cripts, 2012-2015. [4] Donald. E. Knuth. Review of History of Binary and Other Nondecimal Numeration, by Anton Glaser. Historia Math- ematica 10:2 (1983), 236-243. [5] Francis Lieber, E. Wigglesworth, and T. G. Bradford, editors. Encyclopaedia Americana: A Popular Dictionary of Arts, Sciences, Literature, History, Politics and Biography. Vol. IX. B. B. Mussey & Co, 1854. [6] F. V. Morley. Thomas Hariot--1560-1621. Scientic Monthly 14:1 (1922), 60-66. [7] John William Shirley. Binary numeration before Leibniz. American Journal of Physics 19:8 (1951), 452-454. [8] Lloyd Strickland. Leibniz on number systems. In Hand- book of the History and Philosophy of Mathematical Practice, edited by Bharath Sriraman. Springer, 2023. [9] Lloyd Strickland and Harry Lewis. Leibniz on Binary: The Invention of Computer Arithmetic. MIT Press, 2022. [dAAAAAASUV] [?] The Mathematical Intelligencer 6 [10] Heinrich Wieleitner and Anton von Braunmuhl. Geschichte der mathematik. T.2, von Cartesius bis zur Wende des 18. Jahrhunderts; von Heinrich Wieleitner; bearbeitet unter Benutzung des Nachlasses von Anton von Braunmuhl. Halfte 1, Arithmetik, Algebra, Analysis. G. J. Goschen, 1911. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aliations. LloydStrickland, Department ofHistory, Politics, andPhilosophy, Manchester Metropolitan University, ManchesterM156BH, UK. E-mail: l.strickland@mmu.ac.uk Please enable JavaScript to view the comments powered by Disqus. Discussion #### TL;DR The binary system, which is a system of representing numbers using only two digits, 0 and 1, is often attributed to Gottfried Leibniz. This paper presents evidence that the system was independently developed by Thomas Harriot around 1604 nearly one century before Leibniz. Harriot's invention was motivated by his work on weighing experiments. In these experiments, Harriot was interested in determining the specific gravity of different substances. Specific gravity is a measure of the density of a substance relative to the density of water. To determine specific gravity, Harriot needed to be able to record weights in very small increments. The traditional method of recording weights in Roman numerals was not precise enough for this task. In this paper the author present's that: - Harriot used the digits 0 and 1 to represent all numbers. - Harriot developed a system of binary arithmetic, which allowed for the addition, subtraction, multiplication, and division of binary numbers. - Harriot also developed a system of binary notation, which allowed for the representation of numbers in binary form. > ***"Although Harriot rightly deserves the accolade of inventing binary many decades before Leibniz, his work on it remained unknown until 1922, and so did not influence Leibniz or anyone else, nor did it play any part in the adoption of binary as computer arithmetic in the 1930s. That is one accolade that still belongs to Leibniz."*** ![Harriot_binary](https:/ /i.imgur.com/WfHg4Ls.png) You can find more pages of Harriot's manuscripts here: [Manuscripts: Thomas Harriot](https:// echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/AYB35Z4D /index.meta&start=480&viewMode=text_image&pn=488) > ***"Although Harriot rightly deserves the accolade of inventing binary many decades before Leibniz, his work on it remained unknown until 1922, and so did not influence Leibniz or anyone else, nor did it play any part in the adoption of binary as computer arithmetic in the 1930s. That is one accolade that still belongs to Leibniz."*** The troy system of measurement is a system of weights that is used to measure precious metals and gemstones. It is based on the grain, which is equal to 64.79891 milligrams. A troy ounce is equal to 480 grains, and a troy pound is equal to 12 troy ounces. Learn more here: [Troy Weight](https://en.wikipedia.org/wiki/Troy_weight) Thomas Harriot was a 16th-century polymath who made significant contributions to mathematics, astronomy, navigation, and optics. He was also one of the first Europeans to use a telescope and to make detailed observations of the Moon. Learn more here: [Thomas Harriot](https:// en.wikipedia.org/wiki/Thomas_Harriot) ![harriot](https:// upload.wikimedia.org/wikipedia/commons/e/e6/ThomasHarriot.jpg) You can learn more about Leibniz's work on binary in this extensive and fascinating article by Stephen Wolfram: [Dropping In on Gottfried Leibniz](https://writings.stephenwolfram.com/2013/05/ dropping-in-on-gottfried-leibniz/) FERMAT'S LIBRARY * team@fermatslibrary.com 1160 Mission Street San Francisco California Products * Margins * Librarian * Journal Club Project * About * Store