Hi Nick.
This is very easy - either the book contains an error or you are misreading it. The syllogism you present is valid: if A is B and B is C then A is indeed C. This can be shown as an Euler diagram as three circles, the smallest inside the next inside the largest.
However, A is not (as you call it) the major term - it is the minor term. B is the middle term and C is the major term. The minor term (A) always ends up as subject of the conclusion, and the major term (C) always ends up as predicate of the conclusion.
The syllogism you present here is called a first figure mood: it has A as subject of the minor premise (all A are B) and C as predicate of the major premise (all B are C). Note in passing that traditionally the major premise is stated first and the minor premise is stated second (while you have stated them in a more 'natural' order) - this is significant insofar as first figure might be confused with fourth figure.
The issue of figures is important because the terms' positions in the premises vary (whence four figures), whereas the conclusion always conventionally has the terms A as subject and C as predicate. The middle term B does not enter it, merely serving in the premises as intermediary between the two others called the extremes.
See Future Logic part I, chapters 8-10, on all this.

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