What is the definition of "horse" in Hard Times?
Thomas Gradgrind wants nothing but facts:
"NOW, what I want is, Facts. Teach these boys and girls nothing but Facts."
When he asks Girl #20 to tell the class what a horse is, she cannot come up with the kind of answer that satisfies the teacher. In fact, Sissy seems to think it is so obvious what a horse is that there should be no need to explain it in words. A horse is a horse. Gradgrind finally announces his disapproval to the class:
"Girl number twenty unable to define a horse!" said Mr. Gradgrind, for the general behoof of all the little pitchers. "Girl number twenty possessed of no facts, in reference to one of the commonest of animals! Some boy's definition of a horse. Bitzer, yours."
Gradgrind refers to his pupils as "little pitchers" because he thinks of them as having empty minds that need to be filled up with facts and more facts. The irony in this situation is that "Girl number twenty" is Sissy Jupe, who works with her father in the circus and has grown up among horses. She knows more about horses than Gradgrind ever will. In fact, she is an accomplished bareback rider. Yet it is Bitzer, Gradgrind's star pupil, a boy who knows no more about horses than his teacher, who comes up with the kind of answer Gradgrind wants to hear.
"Quadruped. Graminivorous. Forty teeth, namely twenty-four grinders, four eye-teeth, and twelve incisive. Sheds coat in the spring; in marshy countries, sheds hoofs, too. Hoofs hard, but requiring to be shod with iron. Age known by marks in mouth."
Obviously Bitzer has only memorized what he has read in a book and is repeating the words like a parrot without even understanding some of what he is parroting. For example, Bitzer probably does not understand the words "quadruped" or "graminivorous."
Dickens draws a comparison between real knowledge and what used to be called "book-learning." Dickens himself did not have a great deal of schooling because he had to go to work while he was only twelve years old. Yet he became one of the world's greatest writers through self-education which included extensive reading.
I presume you're asking about the scene in chapter 2 where Thomas Gradgrind asks the students for the definition of "horse." He calls on a girl first, whom he calls Girl 20, but she is unable to give him an answer. Then he asks for a boy to give him the answer:
"Bitzer," said Thomas Gradgrind. "Your definition of a horse."
"Quadruped. Graminivorous. Forty teeth, namely twenty-four grinders, four eye-teeth, and twelve incisive. Sheds coat in the spring; in marshy countries, sheds hoofs, too. Hoofs hard, but requiring to be shod with iron. Age known by marks in mouth." Thus (and much more) Bitzer.
"Now girl number twenty," said Mr. Gradgrind. "You know what a horse is."
Is Dickens saying that boys are smarter than girls? Does it mean anything that the girl is known only by a number, but the boy's name is used? You decide.