The Virginia Gazette

To the PUBLIC.

Very natural but earnest thought, and that occurred on the subject, a considerable number of years ago, led me directly to a discovery of what, I was soon after satisfied, must be a solution of that problem, by which the circle was to be measured. My researches for a demonstration, where all the difficulty lay, have had their desired effect, by the following, and some other discoveries.

1st. A square has the same proportion to its inscribed circle, as that circle has to a square whose perimeter is equal to its circumference.

A professor, in a university in Europe, having read as above, stopped to observe that it could not be demonstrated; before, however, it came to his turn to make observations, a Gentleman of undoubted abilities had approved of it.

2d. In a circle, the rectangle contained under the sides of the inscribed and circumscribed squares is equal to the inscribed octagon.

These two have been published.

3d. The double sector, a new mathematical instrument, which measures heights and distances by inspection, without calculation.

A description of this was read before the Royal Society in London.

4th. If a cube is cut parallel to its base, at a certain altitude, the sections will be parallelepipeds, and the greater of them equal to the sphere inscribed in the cube.

5th. The rectangle contained under half the circumference of a circle and another certain line is equal to the square of the diameter.

When this proposition is announced in its proper dress, and the relation of the certain line made known, a Geometrician, who understands Euclid's 6th book, can be at no loss for the demonstration, which is direct, easy, and short.

6th. The diameter and circumference of a circle are to each other in the ratio of number to number and commensurable.

The first part of this has an indirect demonstration, like that used by Euclid II. 10.

7th. If the cylinder inscribed in a cube is continued till its altitude is equal to a certain line, then the cylinder and cube will be equal.

Several years ago, a Gentleman of distinguished character and rank, as a Mathematician, in Britain, being applied to, and having heard what I laid before him on this subject, gave for answer, that, if I could prove the circumference of a circle to be a number, those numbers pointed out by my demonstration would express the proportion which the diameter of a circle has to its circumference.

This proof is completed; and I am ready to produce the demonstrations for examination, upon terms that may easily be agreed on.

ANDREW MARR.