The kaleidoscope

by David Brewster

“So likewise a pentagon may be perfectly represented, by finding the fifth part of a circle, and placing the glasses upon the outlines of it; and the fourth part of a circle will likewise produce a square, by means of the glasses, or, by the same rule, will give us any figure of equal sides. I easily suppose that a curious person, by a little practice with these glasses, may make many improvements with them, which, perhaps, I may not have yet discovered, or have, for brevity sake, omitted to describe.

“It next follows that I explain how, by these glasses, we may, from the figure of a circle, drawn upon paper, make an oval; and also, by the same rule, represent a long square from a perfect square. To do this, open the glasses, and fix them to an exact square; place them over a circle, and move them to and fro till you see the representation of the oval figure you like best; and so, having the glasses fixed, in like manner move them over a square piece of work till you find the figure you desire of a long square. In these trials you will meet with many varieties of designs. As for instance, [Fig. 56], although it seems to contain but a confused representation, may be varied into above two hundred different representations, by moving the glasses over it, which are opened and fixed to an exact square. In a word, from the most trifling designs, we may, by this means, produce some thousands of good draughts.

“But that [Fig. 56] may yet be more intelligible and useful, I have drawn on every side of it a scale, divided into equal parts, by which means we may ascertain the just proportion of any design we shall meet within it.

“I have also marked every side of it with a letter, as A, B, C, D, the better to inform my reader of the use of the invention, and put him in the way to find out every design contained in that figure.

“Example I.—Turn the side A to any certain point, either to the north, or to the window of your room; and when you have opened your glasses to an exact square, set one of them on the line of the side D, and the other on the line of the side C, you will then have a square figure four times as big as the engraved design in the plate: but if that representation should not be agreeable, move the glasses (still open to a square) to the number 5 of the side D, so will one of them be parallel to D, and the other stand upon the line of the side C, your first design will then be varied; and so by moving your glasses, in like manner, from point to point, the draughts will differ by every variation of the glasses, till you have discovered at least fifty plans, differing from one another.

“Example II.—Turn the side marked B, of [Fig. 56], to the same point where A was before, and by moving your glasses as you did in the former example, you will discover as great a variety of designs as had been observed in the foregoing experiment; then turn the side C to the place of B, and, managing the glasses in the manner I have directed in the first example, you may have a great variety of different plans, which were not in the former trials; and the fourth side, D, must be managed in the same manner with the others; so that from one plan alone, not exceeding the bigness of a man’s hand, we may vary the figure at least two hundred times; and so, consequently, from five figures of the like nature, we might show about a thousand several sorts of garden-plats; and if it should happen that the reader has any number of plans for parterres or wilderness-works by him, he may, by this method, alter them at his pleasure, and produce such innumerable varieties, that it is not possible the most able designer could ever have contrived.”

In reading the preceding description, the following conclusions cannot fail to be drawn by every person who understands it.

1. Dr. Bradley, like Kircher, considers his mirrors as applicable to regular figures, such as are represented in [Fig. 55], and was entirely unacquainted with the fact, that the inclination of the mirrors must be an even aliquot part of a circle. This is obvious, from his stating that the mirrors may be set at the third, fourth, fifth, sixth, seventh, or eighth part of a circle; for if he had tried to set an irregular object between the mirrors when placed at the third, fifth, or seventh part of 360°, he would have found that a complete figure could not possibly be produced.

2. From the erroneous position of the eye in front of the mirrors, there is such an inequality of light in the reflected sectors, that the last is scarcely visible, and therefore cannot be united into a uniform picture with the real objects.

3. As the place of the eye in Bradley’s instrument is in front, and therefore much nearer the object, or sector, seen by direct vision, the angular magnitudes of all the different sectors are different, and hence they cannot unite into a symmetrical figure. This is so unavoidable a result of the erroneous position of the eye—a position too, rendered necessary from the absurd form of the mirrors—that Bradley actually employs his mirrors to convert a circle into an ellipse, and a square into a rectangle!