one are two?”—­“Twice two is four,” or, “Twice two are four?”—­“Thrice one is or are, three?”—­“Three times one is, or are, three?”—­“Three times naught is, or are, naught?”—­“Thrice three is, or are, nine?”—­“Three times four is, or are, twelve?”—­“Seven times three make, or makes, twenty-one?”—­“Three times his age do not, or does not, equal mine?”—­“Three times the quantity is not, or are not, sufficient?”—­“Three quarters of the men were discharged; 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