Harriot, Thomas - Mss. 6785

281141

[Commentary:

Harriot refers at the end to Euclid, .
If three straight lines be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes be equal to the square on the mean, the three straight lines will be proportional. ]

Of three magnitudes in continuall proportion: the first being given & the
summe of the second & third; to find the second &

Let

a b

be the first & the aggregate of the second & third

b c

. Unto the
line

b c

adde a line æquall which let be

c d

: & unto them both add half of
the first magnitude which let be

e b

. Then let

e f

be perpendicular to

e b E & æ q u a l l

& accomplishe the parallellogramme

e f g d

. Betwixt the lines

e f

f g

or
which is all one

a e

e d

get a mean proportionall which let be

e i

.
It is done by dividing the whole line

a d

into two æquall partes in the poynt

h

& according to the distance

h a

h d

describing the circle

a i d

. & then producing
theline

e f

to the periphery at

i

e i

is the meane proportionall befour [???] of

e f

which is æquall to half of the first magnitude subtrate from

e i

& then

f i

will remayne: & unto

f i

take a line æquall

b k

. Then I say that

b k

is the second proportionall &

k c

the third. Which is thus proved.
first accomplish the square

e i h k

& form the poynt

b

draw the line

b o

parallel to

e i

. the figures

e f m b

m n l o

wilbe squares & their complements

b n

m i

wilbe æquall. Let

n q

be æquall to

k n

& the parallelogramm accomplished

k q p c

. and from

h

let there be drawne the line

h r

parallel to

c p

k q

. Befour I proceede farther I
say that the line

h c

is æquall to

a e

e b

. which is manifest by this lemma. If
there be two magnitudes unæquall: the difference betwixt of there halfes is the difference
betwixt half the difference of the two wholes.
The whole lines are

a b

b d

. their difference
is

a b

. The half lines

h d

and

c d

are the halfes &

h c

their difference, therefour
by the lemma

h c

is is æquall to the half of

a b

that is

e b

. & therefour also

h t

s p

are squares & æquall, & either of them æquall to

e m

c

divides the line

b d

ito two æquall partes, therefour the parallelogramms

b t

c g

are æquall: & let

c z

be æquall to

k t

. Therefour

k p

m l

are
æquall; but

k p

is an oblonge made of

k q

æquall to

a b

the first magnitude
& of

q p

æquall to

k c

the last magnitude; which oblonge or parallelogram
when it is æquall to the square of the middest which is

m l

the square
of

m n

, æquall to

b k

the middest: therefour (by the 17 of the 6th)

a b

b k

k c

are three lines in continuall proportion which was required to be proved.

161 (81)	162 (81v)
163 (82)	164 (82v)
165 (83)	166 (83v)
167 (84)	168 (84v)
169 (85)	170 (85v)

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