Willis . — The Distribution of Species in New Zealand. 443 
length of time it reaches the ends of the islands (o and 1,000). At half 
time it will be at 350 and 750, and so on. 
As w spreads, it is supposed to give rise to endemics, and the actual 
facts of the flora show that in most cases it does so. For this illustration 
I have imagined these to be formed in increasing numbers as time goes on 
and the wides occupy a larger and larger area, but there is no absolute 
necessity for this assumption. In actual fact also, probably some of the 
older — or perhaps even the younger — endemics will themselves give rise to 
other endemics, but as to such possibilities we have absolutely no informa- 
tion whatever, though now that I have shown that some of the phenomena 
that follow evolution can be studied arithmetically, 1 we may expect to gain 
a good deal of new information in such respects. I have, then, imagined all 
the endemics to arise directly from the wides, and to arise only 2 at the levels 
where that wide has reached a dispersal of 200, 400, 600, or 800 miles along 
the islands. I have further imagined 2 that at each of these levels the 
number of endemics will be proportional to the distribution of the wides, so 
that we shall get 1, 2, 3, and 4 arising. I then obtained the actual spots of 
origin by drawing numbers from a hat, when there turned up (a) 520, 
(1 b ) 400, 640, ( c ) 56 o, 460, 360, and (d) 580, 400, 600, 360. Placing a dot at 
each of these points to represent the origin of these species, and drawing 
for each of them a triangle similar to that constructed for the wides, there 
appeared the result given in the diagram. The points at which the 
triangles meet the base line obviously represent the distances along the 
islands reached by these imaginary endemics when the imaginary wide has 
reached the ends of the islands ; thus endemic No. 1 will have reached 120 
and 920 at that time. 
If now we divide New Zealand into ten zones, as is done in the 
diagram by drawing vertical lines at every hundred miles, and then count 
the number of triangles of endemics cut by each of these lines, we shall get 
the number of endemics that occur in that zone. In the particular case in 
hand, for instance, these numbers are 
13688*7522 
or if we take the number of endemics in the whole zone of 100 miles, 
we get 
C5358987322 
In other words, assuming that the wides entered more or less at some 
portion (not the whole) of the islands, and that the endemics were developed 
casually, then the number of endemics to be expected in any zone, starting 
1 By plotting series of triangles or cones like those in the diagram from the actual figures for 
the species of any genus or family, interesting results may be obtained, which will form the subject 
of later papers. 
2 There is no absolute necessity for either of these assumptions ; they are employed simply for 
convenience. 
