DISTANCE OF A SHIP AT SEA
133
from this objection, and depending equally directly on Eucl.
I. 26. If the observer was placed on the top of a tower, lie
had only to use a rough instrument made of a straight stick
and a cross-piece fastened to it so as to be capable of turning
about the fastening (say a nail) so that it could form any
angle with the stick and would remain where it was put.
Then the natural thing would be to fix the stick upright (by
means of a plumb-line) and direct the cross-piece towards the
ship. Next, leaving the cross-piece at the angle so found,
he would turn the stick round, while keeping it vertical, until
the cross-piece pointed to some visible object on the shore,
which would be mentally noted ; after this it would only
be necessary to measure the distance of the object from the
foot of the tower, which distance would, by Eucl. I. 26, be
equal to the distance of the ship. It appears that this precise
method is found in so many practical geometries of the first
century of printing that it must be assumed to have long
been a common expedient. There is a story that one of
Napoleon’s engineers won the Imperial favour by quickly
measuring, in precisely this way, the width of a stream that
blocked the progress of the army. 1
There is even more difficulty about the dictum of Pamphile
implying that Thales first discovered the fact that the angle
in a semicircle is a right angle. Pamphile lived in the reign
of Nero (a. D. 54-68), and is therefore a late authority. The
date of Apollodorus the ‘calculator’ or arithmetician is not
known, but he is given as only one of several authorities who
attributed the proposition to Pythagoras. Again, the story
of the sacrifice of an ox by Thales on the occasion of his
discovery is suspiciously like that told in the distich of
Apollodorus ‘when Pythagoras discovered that famous pro
position, on the strength of which he offered a splendid
sacrifice of oxen ’. But, in quoting the distich of Apollodorus,
Plutarch expresses doubt whether the discovery so celebrated
was that of the theorem of the square of the hypotenuse or
the solution of the problem of ‘ application of areas 3 2 ; there
is nothing about the discovery of the fact of the angle in
a semicircle being a right angle. It may therefore be that
1 David Eugene Smith, The Teaching of Geometry, pp. 172-3.
2 Plutarch, Non posse suaviter vim secundum Epicurum, c. 11, p. 1094 b.