How were elements arranged in the early 1800s?

With great difficulty! Not only had only a few elements been discovered but their atomic weights had not been measured accurately. This entailed a great deal of careful purification and identification by chemists before attempts could be made to arrange them. There were notable efforts by chemists such as Newlands (The Law of Octaves) and Dobereiner (Triads) to arrange the elements in order either by properties or atomic weights or a mixture of the two, but all had flaws.