The English works of Thomas Hobbes of Malmesbury, Volume 07 (of 11) — Thomas Hobbes — John Shaqi
The English works of Thomas Hobbes of Malmesbury, Volume 07 (of 11)
Thomas Hobbes · en
2. That if a straight line be equal to the arc B L, and one end in B,
the other will be somewhere in I C, and higher than the point L.
3. That wheresoever it be, two-thirds of it must be equal to the arc B
K, and one-fifth to the arc K L.
4. That the arc of a quadrant described in the third part of the radius,
or of E G, is equal to the third part of the arc B D, viz. to the arc B
K. I may therefore call a third part of E G, the radius of B K; and a
sixth part of E G, the radius of the arc K L, &c.
5. And lastly, that any straight line drawn from B to I C, if it be
equal to the arc B L, it must cut the half radius I G, whose quadrantal
arc is B L, into the proportion of two to one. For as the whole arc to
the whole E G, so are the parts of it to the parts of E G.
These premises granted, which I think cannot be denied, I say again,
that the straight line B M is equal to the arc B L.
DEMONSTRATION.
[Illustration]
Because B I is to I M, by construction, as two to one, and the line I G
divides the angle B I C in the midst, B _a_ will be to _a_ M as two to
one, that is to say, as the arc B K to the arc K L. From the point M to
the side B C erect a perpendicular M N. And because C M is half C I, the
line M N will be half G C; and B N will be three-quarters of B C; and
the square of B M equal to ten squares of a quarter of B C; and because
B M is to B _a_ as three to two, M N will be to _a_ G as three to two.
But M N is a quarter of E G, therefore _a_ G is two-thirds of a quarter
of E G; that is, one-third of I G; that is, one-sixth of the whole E G.
And I _a_ one-third of E G. Therefore I _a_ is the radius of the arc B
K; and _a_ G the radius of the arc K L; and E G the radius of the whole
arc B L D. Lastly, if a straight line be drawn from B to any other point
of the line I C, though any line may be divided into the proportion of
two to one, it shall not pass through the point _a_, and therefore not
divide the radius of B L, which is I G, into the proportion of two to
one. Therefore no straight line can be drawn from B to I C, except B M,
so as to be equal to the arc B L. Therefore the straight line B M and
the arc B L are equal.
Hence it follows, that seeing the square of B M is equal to ten squares
of a quarter of B C, that a straight line equal to the quadrantal arc B
L D is equal to ten squares of half the radius, as I have divers ways
demonstrated heretofore.
SIX LESSONS
TO THE
PROFESSORS OF THE MATHEMATICS,
ONE OF GEOMETRY, THE OTHER OF ASTRONOMY,
IN THE CHAIRS SET UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE, IN THE
UNIVERSITY OF OXFORD.
TO THE RIGHT HONOURABLE
HENRY LORD PIERREPONT,
VISCOUNT NEWARK, EARL OF KINGSTON, AND
MARQUIS OF DORCHESTER.
MY MOST NOBLE LORD,