"To sleep like a child" implies that tranquil unbroken sleep which parents almost always expect from their children. But sleep disorders are so common in children that almost everyone who has written about their health from Hippocrates to Spock has included a discussion of sleep in their writings.
Attitudes toward sleep, like everything else, change with time. Perhaps no period was more dogmatic about a child's sleep than the mid 1850's in this country. The article below is a good example of what parents of that period read in a popular health journal:
Many a bright and beautiful child is destroyed or made idiotic for life by their nurses, in one of two ways.
By the administration of laudanum, paregoric, opium, or other form of anodyne.
By teaching self-abuse, in order that the exhaustion it produces should promote sleep.
Medical books abound in cases of this lamentable character. How to guard against them with most efficacy, is worthy of inquiry.
All children, under five years of age, will be made the better, healthier, happier, and more good natured, by an undisturbed sleep on one or two hours in the forenoon.
Children, under eighteen months, may require two day naps in summer time.
If a child is regularly put to sleep at the same time, for only three or four days in succession, the habit will so rapidly grow upon it, that with the aid of quiet, and a little darkening of the room, it will, if well, fall to sleep within a few minutes of the time, for weeks and months in succession–such is Nature's love for system and regularity.