Harriot, Thomas, Mss. 6784

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Arithmetica Exegesis
radij by
[Translation: Arithmetical exegesis, for radius by.
Datorum circulorum radij dati
sunt, et centrorum distantiæ
Ergo lateri trianguli z, p, a,
cum sit, ut ah,hp:af,fp.
datur fp. et fh cui æqualis
fb
[Translation: The radii of given circles are given, and the distances of their centres.
Therefore the sides of the triangle zpa, and since ah:hp=af:fp, there is given fp, and fh, which is equal to the angent fb.
Ex fb et bz datis, datur fz.
Sunt igitur duo triangula
datorum laterum fbz, fpz.
constituuntur super eandem
basim fz. datur igitur verti-
cum distantia pb
[Translation: From fb and bz, given, there is given fz.
Therefore there are two triangles with given sides fbz, fpz, constructed on the same base fz.
Therefore the vertical distance pb is given.
Ex triangulo bpz datorum laterum
datur zn et pn perpendicularis
nota igitur bn.
fiunt nη et nλ, æquales radio
circuli circa p.
Dantur, igitur bη et bλ.
Tum:
Datur igitur bc, cuius dimidium
by, radius
[Translation: From the triangle bpz with given sides there is given zn, and the perpendicular pn is known, therefore bn.
There are constructed nη and nλ, equal to the radius of the circle about p.
Therefore there are given bη and bλ.
Then:
Therefore there is given bc, whose half, by, is the sought radius.
Per Canonem triangulorum
alia methodo ut covenit, operatio fit

[Translation: By the Canons for triangles, there is another method, as convenient, which may be carried ore ]
Nota.
per puncta η et λ
fit etiam geometrica
constructio, loco q, l
[Translation: Note.
Through the points η and λ there may also be carried out a geometric construction, instead of q and l.
Arithmetica exegesis
radij ah
cæteris
[Translation: Arithmetical exegesis, for radius ah, given the rest.
Datorum circulorum radij dati
sunt, et centrorum distantiæ
Ergo lateri trianguli z, p, y,
Datur igitur perpendicularis
pn, et linea zn. Unde nota
fit bp.
Cum data pn et po
unde data bo.
Tum, trianguli bpo latera sunt
nota; unde nota perpendicularis
pu. Et linea ou, cui æqualis uh.
Dantur igitur oh et bh.
Dantur igitur hf et pf.
Denique fiat:
Datur igiture ah, quod

[Translation: The radii of given circles are given, and the distances of their centres.
Therefore the sides of the triangle zpy.
Therefore there is given the perpendicular pn, and the line zn. Whence there is known bp.
Since pn and po are given, there is given bo.
Then the sides of triangle bpo are known, whence the perpendicular pu is known. And the line ou, which is equal to uh.
Therefore there are given oh and bh.
Thereofre there are given hf and pf.
Then let there be constructed:
Therefore there is given ah, which was sought.
Geometria exegesis
ipsius radii ah
[Translation: Geometric exegesis, for the same radius ah.
Trium datorum circulorum
centra z, p, y, connectantur.
per z, y fit bc acta
fb faciat angulos rectos cum bc.
Ita pn; quæ secabit circulum
circa p, in puncto o.
Agatur bo, quæ producta secabit
eandem circulum circa p, in h.
Agatur hp et producatur ad
utraque partes quæ secabit bf
in puncto f.
Tum fiat:
Datur igiture ha, et centrum circuli

[Translation: Let the centres of the given circles, z, p, y, be connected.
Through z, y let bc be constructed; fb makes a right angle with bc.
Thus pn, which cuts the circle about p in the point o.
Let there be constructed bo, which extended sill cut the same circle about p at h.
Let hp be constructed and extended on both sides, which will cut bf in the point f.
Then:
Therefore there is given ha, and the centre of the circle sought.

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