Harriot, Thomas - Mss. 6782

533267

[Commentary:

In modern notation, binomes are numbers of the form

\sqrt{m} + \sqrt{n}

where

m

and

n

are integers.
Book X; Definitions , Euclid defined six kinds of binomes, according to various relationships of

m

n

, which for Euclid were geometric lengths. In modern notation, the six binomes may be defined as follows.
Binome 1: a binome of the form

m + \sqrt{n}

with

m > \sqrt{n}

, and

m^{2} = n + k

where

\frac{m}{\sqrt{k}}

is rational; for example

7 + \sqrt{48}

.
Binome 2: a binome of the form

\sqrt{m} + n

with

\sqrt{m} > n

, and

m = n^{2} + k

where

\frac{\sqrt{m}}{\sqrt{k}}

is rational; for example

\sqrt{12} + 3

.
Binome 3: a binome of the form

\sqrt{m} + \sqrt{n}

with

\sqrt{m} > \sqrt{n}

, and

m = n + k

where

\frac{\sqrt{m}}{\sqrt{k}}

is rational; for example

\sqrt{8} + \sqrt{6}

.
Binome 4: a binome of the form

m + \sqrt{n}

with

m > \sqrt{n}

, and

m^{2} = n + k

where

\frac{m}{\sqrt{k}}

is non-rational; for example

2 + \sqrt{2}

.
Binome 5: a binome of the form

\sqrt{m} + n

with

\sqrt{m} > n

, and

m = n^{2} + k

where

\frac{\sqrt{m}}{\sqrt{k}}

is non-rational; for example

\sqrt{2} + 1

.
Binome 6: a binome of the form

\sqrt{m} + \sqrt{n}

with

\sqrt{m} > \sqrt{n}

, and

m = n + k

where

\frac{\sqrt{m}}{\sqrt{k}}

is non-rational; for example

\sqrt{3} + \sqrt{2}

.
Harriot made two further distinctions for binomes of the fifth and sixth kind according to

k

is itself a square (type i) or not (type (ii).
In this and the following folio, Add MS f. , Harriot shows that the square of any binome is always a binome of the first kind. This folio shows his working for first, second, and third binomes. ]

Binomiorum quadrata, sunt binomia
[Translation: Squares of binomes are binomes of the first ]

1.
[Translation: A binomial of the first ]

ut
[Translation: as ]

511 (256)	512 (256v)
513 (257)	514 (257v)
515 (258)	516 (258v)
517 (259)	518 (259v)
519 (260)	520 (260v)

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