*Kigenkai* is a mathematical investigation on circles and spheres. In particular, the volume of a sphere is derived, a value for \(\pi\) estimated, and arc length discussed. The text has been modified and amended by many generations of Japanese scholars, but it is often attributed to Seki Takakazu (also known as Seki Kowa, c.1642-1708), the leading mathematician of the Edo Period (1603-1867). The title page begins the text:

The volume of a sphere is obtained through “slicing”:

The area of a circle is obtained through a process of geometric iteration from which a value for \(\pi\) is obtained. This method is the same as that used by the Chinese mathematician Liu Hui several centuries prior:

For more of the mathematics of Seki Takakazu (also known as Seki Kowa), see Mathematical Treasures - Seki Kowa's *Essentials of Mathematics, *by Frank J. Swetz and Victor J. Katz.

*The images above are presented through the kind cooperation of the National Diet Library, Japan, and are used with permission. They were obtained from the library’s digital gallery exhibit, Japanese Mathematics in the Edo Period, where a complete examination and viewing of the item presented above can be found. *