as “the grammatical subject” with which the verb must agree.  But, among these latter expounders, there are two opposite opinions on the very essential point, whether this “entire expression” requires a singular verb or a plural one:—­as, whether we ought to say, “Twice one is two,” or, “Twice one are two;”—­“Twice two is four,” or, “Twice two are four;”—­“Three times one is three,” or, “Three times one are three;”—­“Three times three is nine,” or, “Three times three are nine.”  Others, again, according to Dr. Bullions, and possibly according to their own notion, find the grammatical subject, sometimes, if not generally, in the multiplying term only; as, perhaps, is the case with those who write or speak as follows:  “If we say, ‘Three times one are three,’ we make ‘times’ the subject of the verb.”—­Bullions, Analyt. and Pract.  Gram., 1849, p. 39.  “Thus, 2 times 1 are 2; 2 times 2 are four; 2 times 3 are 6.”—­Chase’s C. S. Arith., p. 43.  “Say, 2 times O are O; 2 times 1 are 2.”—­Robinson’s American Arith., 1825, p. 24.

OBS. 17.—­Dr. Bullions, with a strange blunder of some sort in almost every sentence, propounds and defends his opinion on this subject thus:  “Numeral adjectives, being also names of numbers, are often used as nouns, and so have the inflection and construction of nouns:  thus, by twos, by tens, by fifties. Two is an even number.  Twice two is four.  Four is equal to twice two.  In some arithmetics the language employed in the operation of multiplying—­such as ’Twice two are four, twice three are six’—­is incorrect.  It should be, ‘Twice two is four,’ &c.; for the word two is used as a singular noun—­the name of a number.  The adverb ‘twice’ is not in construction with it, and consequently does not make it plural.  The meaning is, ‘The number two taken twice is equal to four.’  For the same reason we should say, ‘Three times two is six,’ because the meaning is, ‘Two taken three times is six.’  If we say, ’Three times one are three,’ we make ‘times’ the subject of the verb, whereas the subject of the verb really is ‘one,’ and ‘times’ is in the objective of number (Sec.828). 2:4::  6:12, should be read, ’As 2 is to 4, so is 6 to 12;’ not ‘As two are to four, so are six to twelve.’  But when numerals denoting more than one, are used as adjectives, with a substantive expressed or understood, they must have a plural construction.”—­Bullions, Analyt. and Pract.  Gram., 1849, p. 39.