This leads to defining the ratio of the circumference to the diameter as [pi]. Although this is a Greek letter, it was not used by the Greeks to represent this ratio. Indeed, it was not until 1706 that an English writer, William Jones, in his "Synopsis Palmariorum Matheseos," used it in this way, it being the initial letter of the Greek word for "periphery." After establishing the properties that _c_ = 2[pi]_r_, and _a_ = [pi]_r_^2, the textbooks follow the Greek custom and proceed to show how to inscribe and circumscribe various regular polygons, the purpose being to use these in computing the approximate numerical value of [pi]. Of these regular polygons two are of special interest, and these will now be considered. PROBLEM. _To inscribe a regular hexagon in a circle._ That the side of a regular inscribed hexagon equals the radius must have been recognized very early. The common divisions of the circle in ancient art are into four, six, and eight equal parts. No draftsman could have worked with a pair of compasses without quickly learning how to effect these divisions, and that compasses were early used is attested by the specimens of these instruments often seen in museums. There is a tradition that the ancient Babylonians considered the circle of the year as made up of 360 days, whence they took the circle as composed of 360 steps or grades (degrees). This tradition is without historic foundation, however, there being no authority in the inscriptions for this assumption of the 360-division by the Babylonians, who seem rather to have preferred 8, 12, 120, 240, and 480 as their division numbers. The story of 360 deg. in the Babylonian circle seems to start with Achilles Tatius, an Alexandrian grammarian of the second or third century A.D. It is possible, however, that the Babylonians got their favorite number 60 (as in 60 seconds make a minute, 60 minutes make an hour or degree) from the hexagon in a circle (1/6 of 360 deg. = 60 deg.), although the probabilities seem to be that there is no such connection.[83] The applications of this problem to mensuration are numerous. The fact that we may use for tiles on a floor three regular polygons--the triangle, square, and hexagon--is noteworthy, a fact that Proclus tells us was recognized by Pythagoras. The measurement of the regular hexagon, given one side, may be used in computing sections of hexagonal columns, in finding areas of flower beds, and in other similar cases. This review of the names of the polygons offers an opportunity to impress their etymology again on the mind. In this case, for example, we have "hexagon" from the Greek words for "six" and "angle." PROBLEM. _To inscribe a regular decagon in a given circle._