https://en.wikipedia.org/wiki/Reynolds_v._United_States
The supreme courts opinion....
http://law2.umkc.edu/faculty/projects/f ... dsvus.html
Basically the SC justice determined that ...
Polygamy has always been odious among the northern and western nations of Europe, and, until the establishment of the Mormon Church, was almost exclusively a feature of the life of Asiatic and of African people. At common law, the second marriage was always void, and from the earliest history of England polygamy has been treated as an offence against society....
From that day to this we think it may safely be said there never has been a time in any State of the Union when polygamy has not been an offence against society, cognizable by the civil courts and punishable with more or less severity. In the face of all this evidence, it is impossible to believe that the constitutional guaranty of religious freedom was intended to prohibit legislation in respect to this most important feature of social life. Marriage, while from its very nature a sacred obligation, is nevertheless, in most civilized nations, a civil contract, and usually regulated by law. Upon it society may be said to be built, and out of its fruits spring social relations and social obligations and duties, with which government is necessarily required to deal. In fact, according as monogamous or polygamous marriages are allowed, do we find the principles on which the government of the people, to a greater or less extent, rests. An exceptional colony of polygamists under an exceptional leadership may sometimes exist for a time without appearing to disturb the social condition of the people who surround it; but there cannot be a doubt that, unless restricted by some form of constitution, it is within the legitimate scope of the power of every civil government to determine whether polygamy or monogamy shall be the law of social life under its dominion.
In our opinion, the statute immediately under consideration is within the legislative power of Congress. It is constitutional and valid as prescribing a rule of action for all those residing in the Territories, and in places over which the United States have exclusive control. This being so, the only question which remains is, whether those who make polygamy a part of their religion are excepted from the operation of the statute. If they are, then those who do not make polygamy a part of their religious belief may be found guilty and punished, while those who do, must be acquitted and go free. This would be introducing a new element into criminal law. Laws are made for the government of actions, and while they cannot interfere with mere religious belief and opinions, they may with practices. Suppose one believed that human sacrifices were a necessary part of religious worship, would it be seriously contended that the civil government under which he lived could not interfere to prevent a sacrifice? Or if a wife religiously believed it was her duty to burn herself upon the funeral pile of her dead husband, would it be beyond the power of the civil government to prevent her carrying her belief into practice?
So here, as a law of the organization of society under the exclusive dominion of the United States, it is provided that plural marriages shall not be allowed. Can a man excuse his practices to the contrary because of his religious belief? The permit this would be to make the professed doctrines of religious belief superior to the law of the land, and in effect to permit every citizen to become a law unto himself. Government could exist only in name under such circumstances....