242 The N.Z. Journal of Science and Technology. [Jan. 
most interesting little skit called 44 Flatland ; or, A 'Romance of Many 
Dimensions,” and, like many another skit, it had a' lot of sober common- 
sense dished up with a most attractive condiment of humour and levity. 
The author, Professor Abbott, pictured a being living in a space of two 
dimensions—a flat plane—and having no experience or conception of the 
third dimension of height. In its own flat world this, being was one of 
the very highest order—he was, in fact, a circle. In “ Flatland ” there 
were beings of all sorts of shapes—triangles, squares, polygons, et hoc genus 
omne —but they were all, as it were/cut out of a thin sheet of paper ; none 
of them had any thickness, or height—the third dimension, as we know 
it, was missing. In this interesting little world the form which had the 
greatest number of sides was the highest organization ; and that is why 
I have described the being pictured by Professor Abbott, which had the 
form of a circle, as of the highest order, for a circle is only a polygon of 
an infinite number of sides. The author of 44 Flatland ” was rude enough 
to state that the ladies of 44 Flatland ” were mere lines—one-sided beings, 
that is to say—and therefore very humble in the scale of development. 
Of course, it is quite unnecessary to say that Professor Abbott could not 
really have had any experience of ladies, not even of ladies of 44 Flatland.” 
Now, this highly organized being of 44 Flatland ”—this circle—knew all 
about the world in which he lived ; but it was a limited world, a world 
in which everything happened in length and breadth, and no one thought 
of up and down. Into this world one day there came a sphere, a being 
of three dimensions—a being as highly organized in his own world as the 
circle was in his. And the two met, and, as highly organized beings (such 
as ourselves) will do, they had an argument. Said the sphere to the 
circle, I am a sphere.” To which the circle replied, 44 You are not; 
you are a circle like myself.” And to the circle this appeared to be so, 
for he could only see and become conscious of that part of the sphere which 
intersected the 44 Flatland ” in which he lived. 44 No,” said the sphere, 
44 1 am a sphere, and I will show you so ”; and he proceeded to penetrate 
himself more deeply into the 44 Flatland ” in which the other lived. 
44 You are still only a circle,” said the circle, 44 though truly the most 
remarkable one I have met, for you seem to expand and contract in a very 
marvellous way ; but, for all that, you are only a circle.” Then the 
sphere decided that the only way to convince the circle that he (the sphere) 
was a different sort of being altogether was to take the circle bodily out 
of 44 Flatland ” and into a world of three dimensions, and he would then 
understand. And so the sphere took the circle out of his flat world, with 
results which were very interesting and amusing to read, but which are 
not important from our point of view. Now, this amusing allegory and 
skit has an almost prophetic bearing on what we have been considering, 
though 44 Flatland ” was written in 1881, and this is 1920. For it was 
the non-recognition by the circle of the third dimension of height that 
caused that gentleman to be astonished, at the sphere’s seemingly extra¬ 
ordinary shrinkings and expandings, which were in reality due to the 
sphere’s penetrating to a greater or less depth into the plane in which he 
lived. And here, again to-day, we have a case of seemingly astonishing 
shortenings and lengthenings in bodies moving with or against the ether 
stream, and it is to Minkowski that we owe the explanation. But I can 
feel that I am up against a considerable difficulty. Minkowski considered 
that space in in reality a four-dimensional entity or continuum, and not, 
as it seems to us to be, one of only three ; and the fourth dimension is not 
time, but what mathematicians call 44 i ” time. It is not possible to render 
