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Morley (1899–1980) noted that an unpublished manuscript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.