Harriot, Thomas - Mss. 6784

395198

6.)

Sint tres dati circuli,

b r d

h u e

b g c

, sese mutuo
contingentes in punctis

b

g

e

,
cuius centra,

z

a

y

[Translation: Let there be three given circles,

b r d

h u e

b g c

, mutually touching in the points

b

g

e

, whose centres are at

a

a

y

Oportet invenire circulum
contingentem tres datos:
(nempe,

r h t

, cius centrum,

p

[Translation: One must find the circle touching the three given ones (that is,

r h t

, with centre

p

Per centra

z

y

, agatur recta
et continuetur ad utraque partes
et fit

b

z

y

d

c

.
Et ad illam lineam

b c

, fit

a m

perpendicularis.
Continuetur ad partes contrarias
usque ad

k

, et fit

s k = s a

.
Tum primo, agatur recta

b s

quæ secabit periferiam circuli
cuius centrum

a

in puncto

h

.
Secundo, agatur recta

b k

quæ secabit

a h

productam in
puncto

p

.
Ultimo, centro

p

, intervallo

p h

describatur circulus.
Dico quod: ille est circulus quæsitus
et contingit tres datos in
punctis,

h

r

t

[Translation: Through the centres

z

and

y

, a line is drawn and continued on both sides, and so there are

b

z

y

d

c

.
And to that line

b c

, let

a m

be perpendicular.
It is continued to both sides as far as

k

, and let

s k = s a

.
Then, first, there is drawn the line

b s

, which will cut the circumference of the circle with centre

a

in the point

h

.
Second, there is drawn the line

b l

, which will cut

a h

extended, in the point

p

.
Finally, with centre

p

and radius

p h

, there is drawn the required circle.
I say that this is the circle sought, and it touches the three given at the points

h

r

t

Exegesis arithmetica
pro

p h

[Translation: Arithmetical exegesis, for radius

p h

Datorum circulorum radii
dati sunt, et centrorum
distantiæ.
Ergo lateri trianguli

z a y

data sunt. Inde perpendicularis

a m

, et recta

d m

. Inde tota

b m

.
Inde datur

b a

. Inde

b θ

.
Tum cum datur

s a

a m

, datur

s m

et inde

b s

. Et cum datur

x b

b θ

, datur

b h

h s

.
Tum lineæ

b c

fit

b f

ad angulos
rectos et

a p

pro-
ducta concurret cum illa
in puncto

f

f b

f h

sunt
æquales. et triangulum

f b h

simile est triangulo

a s h

,
cuius latera data sunt. et
antea datum fuit

b h

. ergo dantur

f b

f h

.
[…]
Ergo tota

f a

datur
[…]
Ergo datur

a p

sed antea nota fuit

a h

,
ergo

h p

datur
Quod
[Translation: The radii of the fiven circles are given, and the distances of their centres.
Therefore the sides of the triangles

z a y

are given. Hence the perpendicular

a m

, and the line

d m

. Hence the total,

b m

. Hence there is given

b a

. Hence

b θ

. Then since

s a

and

a m

are given,

s m

is given and thence

b s

. And since

x b

and

b θ

are given,

b h

and

h s

are given.
Then the line

b c

is at right angles to

b f

and

a p

extended meets with it at the point

f

. The lines

f b

and

f h

are equal. And the triangle

f b h

is similar to triangle

a s h

, whose sides are given. And earlier

b h

was given. Therefore

f b

and

f h

are given.

Therefore the total

f a

is given.

Therefore there is given

a p

, but earlier

a h

became known, therefore

h p

is given.
Which was ]

Per doctrinam sinuum
opus abbreviatur, sed
alia method ut
[Translation: By the doctrine of sines, the work is shorter, but another method, as ]

301 (151)	302 (151v)
303 (152)	304 (152v)
305 (153)	306 (153v)
307 (154)	308 (154v)
309 (155)	310 (155v)

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