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...
Vol.:(0123456789)
The Mathematical Intelligencer
⚫
© 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Morley (1899–1980) noted that an unpublished manuscript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.
⚫
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 × 109 = 11881, 13 × 13 = 169,
and 13 × 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 × 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
× 111 = 100011 (i.e., 5 × 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 × 3 =
9; 7 × 7 = 49; and 45 × 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
[
×
]
2
1
2
[
×
]
2
[
×
]
2
1
[
×
]
1
2
[
×
]
2
[
×
]
2
[
×
]
2
1
[
×
]
1
[
×
]
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.
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, Düsseldorf, 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
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is not permitted by statutory regulation or exceeds the
permitted use, you will need to obtain permission directly
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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. Encyclopædia 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.
⚫
The Mathematical Intelligencer
6
[10] Heinrich Wieleitner and Anton von Braunmühl.
Geschichte der mathematik. T.2, von Cartesius bis zur Wende
des 18. Jahrhunderts; von Heinrich Wieleitner; bearbeitet
unter Benutzung des Nachlasses von Anton von Braunmühl.
Hälfte 1, Arithmetik, Algebra, Analysis. G. J. Göschen,
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
#### 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."***
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)
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/)
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)
