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.