The picture pasted in this page comes from a Scientific American of 1872. An amusing introduction states that:
"It is a noticeable fact that in workshops and factories where numbers of men are employed, a large percentage of the operatives will be found to be prematurely bald. If the cause of this affliction be sought, it will be traced to the pernicious habit of continually wearing the close cap or hat, thus keeping the scalp at an unnatural heat and shutting off from it all ventilation. The hair under such treatment is, as a matter of course, weakend, and, decaying at the roots, falls out in large quantities."
The solution offered to this vexing problem; is a paper cap that is inexpensive, clean, and airy as it is far better to wear this cap than to....
continue wearing a grimy, greasy, thick piece of felt or cloth for months at a time.
So here are the instructions from the article. Grab a sheet of newspaper or brown paper and:
First, provide a sheet of moderately thick brown paper, size from eighteen inches to two feet, shape as in Fig. 1, Smooth it out perfectly flat and double over as in Fig. 2. Turn it, round with the fold from you. and mark the exact middle of the piece at A Fig. 3.
Then bring down both corners and measure off on the edge, B, from the Point, A, Fig. 3, a distance equal to one quarter the circumference of your head. Mark the point. Now, turn the paper over so that the under side will be uppermost, and bend the apex of the triangle back from the point just marked. as in Fig. 4.
Fold over the sides, Figs. 5 and 6. and with scissors cut off the lower portion, C, below the dotted line find also the points of the two lower corners of the pieces just bent over.
Next unfold the paper; spread it out flat. You'll find a square marked in the middle, and creases leading from there to the corners of the paper. Double up the material on these creases, so as to bring up the paper as sides of a box, of which the middle square is the bottom as in Fig-7. Smooth the folds flat and your work will appear as in Figure 8.
Lastly turn up the edges of the box all around twice, folding the paper on itself.
Your cap is then complete and if the measurement directed above was correctly made, it will exactly fit your head.