The achievements of the colonial artists，particularly those of Copley，West, and Peale，lent credence to the boast that the new nation was capable of encouraging genius and that political liberty was congenial to the development of taste——a necessary step before art could assume an important role in the new republic.
(The achievements…lent credence to the boast that…and that…为句子主干，两个that引导的两个从句都作boast的同位语)
这些殖民地艺术家的成就，尤其是考伯莱、韦斯特和皮尔的成就，，在新的共和国里担当重要角色之前所必须经历的一步。
解析：
主句：The achievements of the colonial artists lent credence to the boast
同位语从句1：that the new nation was capable of encouraging genius
同位语从句2：that political liberty was congenial to the development of taste 其中
“——“引导的从句：a necessary step before art could assume an important role in the new republic.
修饰成分：
插入语：，particularly those of Copley，West, and Peale，
其中两个同位语从句1和2由and并列在一起，共同修饰主句中的宾语boast“——“引导的从句是用来修饰整个主句的，对其进行补充描述插入语修饰artists
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