试论梅文鼎球面余弦定理及符号判定法

探寻梅文鼎(1633—1721)《环中黍尺》中的球面余弦定理及符号判定法。《环中黍尺》中所载"加减约法"针对当时余弦值为正而产生的符号改变问题所设,结论正确,但誊写有误。梅文鼎给出数种余弦定理的正确证明法,并指出每种算法在符号判定、数值查询、计算简便等方面的特点。他综合诸法优势得出"加减捷法",由此阐明《测量全义》卷九中算式的数学原理,而该法符号判定及特殊条件下算法变化均正确,戴震"矢较法"实不出梅氏范围。至此,明末传入的球面余弦定理得到圆满解决。     

Ramet population ecology of Panicum virgatum in the field —Competitively random growth of ramets and foraging behavior of ramet populations

The authors investigated the heterogeneous size patterns and dynamic growth of the ramet population of Panicum virgatum, a clonal caespitose plant, limited to the space occupied by a ramet bunch and the time of the ramet yearly life cycle, to understand the ecology of clonal caespitose plants in the field, where the ramet bunch generally consisted of more than one genet. Dynamic life tables for ramet populations were established by the replacement of living ramets at the present time with ＂dead＂ ones in past time. These tables revealed stable coexisting patterns of isometric and allometric growing processes of ramets in mass and height respectively, which approximately followed the historic trajectory of a density-independent population. The ecology of clonal caespitose plants is further discussed based on the competitively random growth of ramet individuals, including the scale of foraging behavior. In the field, the ramet population ecology of switchgrass may be a statistical result of competitively random growth of ramet individuals. The foraging behavior of a ramet population could then be presented as a process in which ramet individuals competed with each other for light and grew randomly, while at the same time a relatively stable dynamic growth pattern was apparent at the level of the ramet population, and the functional leaves were placed properly in time and space.