Use of determiners as adjectives

In a grammar book that I'm reading, an adjective is defined as:

A word that modifies a noun or a pronoun. (To modify is to limit or point out or describe: that book; another chance; the blue ribbon). For convenience the articles a, an, and the are usually classified under adjectives.

When I looked up in dictionary,

a, an, the, that etc. are known as determiners. How can they be classified under adjectives?

The term determiner is newish (about 80 years old), and hasn't had much uptake in school grammars. The study, in English, of French and other modern languages has employed the term DETERMINATIVE ADJECTIVE since at least 1806, when Dufief wrote,

"S. Why do you call them determinative?

M. Because, when they are expressed before nouns, we know how often the object represented by the noun is repeated" (p. 40).

In 1924, Palmer was the first to try to corral this group of theretofore-heterogeneous English words by adopting the concept from the French analysis.

"To group with the pronouns all determinative adjectives (eg article-like, demonstratives, possessives, numerals, etc.), shortening the term to determinatives (the "déterminatifs" of the French grammarians) firstly because there are divergent opinions as to whether they are adjectives or pronouns, and secondly, because most of the members of this category may be used indifferently as pronouns or as modifiers of nouns" (p 24).

And, in 1933, Bloomfield introduced the slightly different term, DETERMINER, into English linguistics when he wrote,

"our limiting adjectives fall into two sub-classes of determiners and numeratives [1]... The determiners are defined by the fact that certain types of noun expressions (such as house or big house) are always accompanied by a determiner (as, this house, a big house)" (p 203).

The following extract from the Cambridge Grammar of the English Language (page 538) explains why the classification of determiners will vary according to the reference you are using:

This book follows the practice of most work in modern linguistics in recognising a primary part-of-speech distinction between adjectives and determinatives. In traditional grammar, by contrast, determinatives are wholly or almost wholly subsumed under the adjective category - they are said to be 'limiting adjectives', as opposed to 'descriptive adjectives'.