The construction of the wonderful canon of logarithms

8
1
Construction of the Canon.
3 S, 4 S, &c., are said to decrease geometrically,
because in equal times they are diminished by
unequal spaces similarly proportioned. Let the
sine T S be represented in numbers by looooooo,
I S by 9000000, 2 S by 8100000, 3 S by 7290000,
4 S by 6561000; then these numbers are said to
decrease geometrically, being diminished in equal
times by a like proportion.
25. Whence a geometrically moving point approaching a
fixed one has its velocities proportionate to its distances
from thefixed one.
Thus, referring to the preceding figure, I say
that when the geometrically moving point G is at
T, its velocity is as the distance T S, and when G
is at I its velocity is as i S, and when at 2 its
velocity is as 2 S, and so of the others. Hence,
whatever be the proportion of the distances T S,
I S, 2 S, 3 S, 4 S, &c., to each other, that of the
velocities of G at the points T, i, 2, 3, 4, &c,, to
one another, will be the same.
For we observe that a moving point is declared
more or less swift, according as it is seen to be
borne over a greater or less space in equal times.
Hence the ratio of the spaces traversed is necessarily
the same as that of the velocities. But the
ratio of the spaces traversed in equal times, T i,
I 2, 2 3, 3 4, 4 5, &c., is that of the distances T S,
I S, 2 S, 3 S, 4 S, &c[*] Hence it follows that
the ratio to one another of the distances of G from
S, namely T S, i S, 2 S, 3 S, 4 S, &c., is the
same as that of the velocities
T, I, 2, 3, 4, &c., respectively.
of G at the points
[*] It is evident that the ratio of the spaces traversed
T I, I 2, 2 3, 3 4, 4 5, .&c., is that of the distances T S,
iS,