As is well known, the most important development in the history of Chinese mathematics is materialized in Song-Yuan era through tianyuanshu up to siyuanshu for constructing equations and zengcheng kaifangfa for solving them. There are only two authors in the period, Li Ye and Zhu Shijie who left works dealing with them. They were almost forgotten until the late 18th century in China but Zhu's Suanxue Qimeng(1299) had been a main reference for the Joseon mathematics. Commentary by Luo Shilin on Zhu's Siyuan Yujian(1303) was brought into Joseon in the mid-19th century which induced a great attention to Joseon mathematicians with a thorough understanding of Zhu's tianyuanshu. We discuss the history that Joseon mathematicians succeeded to obtain the mathematical structures of Siyuan Yujian based on the Zhu's tianyuanshu.

In this paper, we discuss how ancient Egyptians find out the area of the circle based on $\ll$Ahmose Papyrus$\gg$. Vogel and Engels studied the quadrature of the circle, one of the basic concepts of ancient Egyptian mathematics. We look closely at the interpretation based on the approximate right triangle of Robins and Shute. As circumstantial evidence for Robbins and Shute's hypothesis, Egyptians prior to the 12th dynasty considered the perception of a right triangle as examples of 'simultaneous equation', 'unit of length', 'unit of slope', 'Egyptian triple', and 'right triangles transfer to Greece'. Finally, we present a method to utilize the squaring the circle by ancient Egyptians interpreted by Robbins and Shute as the dynamic symmetry of Hambidge.

$Ces{\grave{a}}ro$ summability is a generalized convergence criterion for infinite series. We have investigated the classical results of summability for Fourier series from 1897 to 1957. In this paper, we are concerned with the summability and summation methods for Fourier Series from 1960 to 2010. Many authors have studied the subject during this period. Especially, G.M. Petersen,$K{\hat{o}}si$ Kanno, S.R. Sinha, Fu Cheng Hsiang, Prem Chandra, G. D. Dikshit, B. E. Rhoades and others had studied neoclassical results on the summability of Fourier series from 1960 to 1989. We investigate the results on the summability for Fourier series from 1990 to 2010 in section 3. In conclusion, we present the research minor lineage on summability for Fourier series from 1960 to 2010. $H{\ddot{u}}seyin$ Bor is the earliest researcher on ${\mid}{\bar{N}},p_n{\mid}_k$-summability. Thus we consider his research results and achievements on ${\mid}{\bar{N}},p_n{\mid}_k$-summability and ${\mid}{\bar{N}},p_n,{\gamma}{\mid}_k$-summability.

The goal of this essay is to examine metaphors for mathematics and to discuss philosophical problems related to them. Two metaphors for mathematics are well known. One is a tree and the other is a building. The former was proposed by Pasch, and the latter by Hilbert. The difference between these metaphors comes from different philosophies. Pasch's philosophy is a combination of empiricism and deductivism, and Hilbert's is formalism whose final task is to prove the consistency of mathematics. In this essay, I try to combine two metaphors from the standpoint that 'mathematics is a part of the ecosystem of science', because each of them is not good enough to reflect the holistic mathematics. In order to understand mathematics holistically, I suggest the criteria of the desirable philosophy of mathematics. The criteria consists of three categories: philosophy, history, and practice. According to the criteria, I argue that it is necessary to pay attention to Pasch's philosophy of mathematics as having more explanatory power than Hilbert's, though formalism is the dominant paradigm of modern mathematics. The reason why Pasch's philosophy is more explanatory is that it contains empirical nature. Modern philosophy of mathematics also tends to emphasize the empirical nature, and the synthesis of forms with contents agrees with the ecological analogy for mathematics.