and three quarters of the money was, or were, sent back?"--"As 2 is to 4, so is 6 to 12;" or, "As two are to four, so are six to twelve?"
OBS. 15.--Most of the foregoing expressions, though all are perhaps intelligible enough in common practice, are, in some respect, difficult of analysis, or grammatical resolution. I think it possible, however, to frame an argument of some plausibility in favour of every one of them. Yet it is hardly to be supposed, that any teacher will judge them all to be alike justifiable, or feel no interest in the questions which have been raised about them. That the language of arithmetic is often defective or questionable in respect to grammar, may be seen not only in many an ill choice between the foregoing variant and contrasted modes of expression, but in sundry other examples, of a somewhat similar character, for which it may be less easy to find advocates and published arguments. What critic will not judge the following phraseology to be faulty? "4 times two units is 8 units, and 4 times 5 tens is twenty tens."--Chase's Common School Arithmetic, 1848, p. 42. Or this? "1 time 1 is l. 2 times 1 are 2; 1 time 4 is 4, 2 times 4 are 8."--Ray's Arithmetic, 1853. Or this? "8 and 7 is 15, 9's out leaves 6; 3 and 8 is 11, 9's out leaves 2."--Babcock's Practical Arithmetic, 1829, p. 22. Or this again? "3 times 3 is 9, and 2 we had to carry is 11."--Ib., p. 20.
OBS. 16.--There are several different opinions as to what constitutes the grammatical subject of the verb in any ordinary English expression of multiplication. Besides this, we have some variety in the phraseology which precedes the verb; so that it is by no means certain, either that the multiplying terms are always of the same part of speech, or that the true nominative to the verb is not essentially different in different examples. Some absurdly teach, that an abstract number is necessarily expressed by "a singular noun," with only a singular meaning; that such a number, when multiplied, is always, of itself the subject of the assertion; and, consequently, that the verb must be singular, as agreeing only with this "singular noun." Others, not knowing how to parse separately the multiplying word or words and the number multiplied, take them both or all together 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 (§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.
OBS. 18.--Since nouns and adjectives are different parts of speech, the suggestion, that, "Numeral adjectives are also names, or nouns," is, upon the very face of it, a flat absurdity; and the notion that "the name of a number" above unity, conveys only and always the idea of unity, like an ordinary "singular noun," is an other. A number in arithmetic is most commonly an adjective in grammar; and it is always, in form, an expression that tells how many, or--"denotes how many things are spoken of."--Chase, p. 11. But the name of a number is also a number, whenever it is not made plural in form. Thus four is a number, but fours is not; so ten is a number, but tens is not. Arithmetical numbers, which run on to infinity, severally consist of a definite idea of how many; each is a precise count by the unit; one being the beginning of the series, and the measure of every successive step. Grammatical numbers are only the verbal forms which distinguish one thing from more of the same sort. Thus the word fours or tens, unless some arithmetical number be prefixed to it, signifies nothing but a mere plurality which repeats indefinitely the collective idea of four or ten.
OBS. 19.--All actual names of numbers calculative, except one, (for naught, though it fills a place among numbers, is, in itself, a mere negation of number; and such terms as oneness, unity, duality, are not used in calculation,) are collective nouns--a circumstance which seems to make the discussion of the present topic appropriate to the location which is here given it under Rule 15th. Each of them denotes a particular aggregate of units. And if each, as signifying one whole, may convey the idea of unity, and take a singular verb; each, again, as denoting so many units, may
