"The place was known by the name of the Octagon. The reason why it was called by this name was, that the principal sitting-room in the house was built in the form of an octagon, that is, instead of having four sides, as a room usually has, this room had eight sides. An octagon is a figure of eight sides.
"A figure of four sides is called a square. A figure of five sides is called a pentagon, of six sides a hexagon, of eight sides an octagon. There might be a figure of seven sides, but it would not be very easily made, and it would not be very pretty when it was made, and so it is seldom used or spoken of. But octagons and hexagons are very common, for they are easily made, and they are very regular and symmetrical in form."
The object of all this is, doubtless, to impart valuable information. But while such slipshod writing is singularly uninteresting, it may also be censured as inaccurate. Mr. Abbott seems to think all polygons necessarily regular. Any child can make a heptagon at once, notwithstanding Mr. Abbott calls it so difficult. A _regular_ heptagon, indeed, is another matter. Then what does he mean by saying octagons and hexagons are very regular? A regular octagon is regular, though an octagon in general is no more regular than any other figure. But Mr. Abbott continues:--
"If you wish to see exactly what the form of an octagon is, you can make one in this way. First cut out a piece of paper in the form of a square. This square will, of course, have four sides and four corners. Now, if you cut off the four corners, you will have four new sides, for at every place where you cut off a corner you will have a new side. These four new sides, together with the parts of the old sides that are left, will make eight sides, and so you will have an octagon.
"If you wish your octagon to be regular, you must be careful how much you cut off at each corner. If you cut off too little, the new sides which you make will not be so long as what remains of the old ones. If you cut off too much, they will be longer. You had better cut off a little at first from each corner, all around, and then compare the new sides with what is left of the old ones. You can then cut off a little more, and so on, until you make your octagon nearly regular.