THE ANALYST. Published the First of each Month. Each Number will contain not less than 16 pages large 8vo. TERMS, $2.00 PER YEAR, IN ADVANCE. In the following article I propose to point out some interesting relations that exist between certain lines and areas of polygons, that, so far as I know, have never heretofore been announced; and to present a method for the construction of certain polygons that have never been constructed geometrically, which, though not strictly geometrical, yet from the analogy of the construction to strictly geometrical constructions I am induced to believe that geometrical constructions for these polygons may yet be found. 1. Draw the line A B= the unit of our scale, and bisect it in E, and at E'erect the indefinite perpendicular E F. With A B as radius describe the arc AG, and draw the straight line AC cutting the arc A G in D, so that D C shall equal A B. Then is A B one side of a pentagon inscribed in a circle which passes through the three points A, B and C. 2. Draw the straight line BI cutting the arc A G in H so that HI = A H. Then is A B one side of a heptagon inscribed in a circle which passes through the three points A, B and I. 3. Through K, the point where the arc A G cuts the perpendicular E F, draw the line L M parallel to A B, and draw the straight line A N cutting L M in P so that PN shall equal twice A B. Then is A B one side of a nonagon inscribed in a circle which passes through the three points A, B and N. The demonstration of these constructions is easy and is therefore omitted for the sake of brevity. = It in the heptagon whose sides each equals one, we put the chord of the arc which contains three of the equal sides 2+, then will 2 + x be the radius of a circle in which if a polygon of double the number of sides (14) be described each of these fourteen equal sides will one; and the length of one side of a heptagon described in a circle whose radius is one will be 1 x. Also, if in the nonagon each of whose sides equals one we put the chord of the arc which contains three of the equal sides 2+, then will one side of the nonagon described in a circle whose radius is one be √1 —∞. And in general, if in a polygon of n equal sides (n being any number greater than six) each of which equals one, we put 2+x= the chord of the arc which contains three of the equal sides, I will be the length of one side of a polygon of n sides described in a circle whose radius is one. Let ẞ represent the angle subtended by one of the equal sides of a polygon of n sides. Then is = √2 √1 — cos ẞ=√ 2 V1 — √ 1—sin3 3=√1+sin ẞ — √ 1—sin ß. That is, one side of any regular polygon inscribed in a circle whose radius is one, is represented in functions of the sine of the arc subtended by that side, by or by its development √1 + sin ẞ -- √ 1 — sin ß, sin ẞ + sin3 ẞ + 252 sin3 ẞ + &c. (1) By assigning any value to ß, n will be determined. And if ẞ be taken very small, n times (1) will represent approximately the circumference of a circle whose radius is one. A very interesting relation exists between the isosceles triangles which are formed in the construction of polygons. The triangle A B C of the pentagon is divided into two triangles by the line B D. Multiply the area of the triangle A B D by the line B C and the product equals the area of the triangle B D C. Multiply the area of |