What they are. easily dis not learnt. The class of questions to which I refer, embraces examples like the following: Add of a day, 13 of an hour, and of a second together. It is certainly true that a boy will make marvellous progress in the text-book, if you limit The subject him to those examples in which the fractions posed of, but have a common unit. But, will he ever understand the science of fractions unless his mind be steadily and always turned to the unit of the fraction, as the basis? Will he understand the value of one equal part, so as to compare and unite it with another equal part, unless he first apprehends, clearly, the units from which those parts were derived? Last objection stated. 4th. By placing the Denominate Numbers between Vulgar and Decimal Fractions, the general subject of fractional arithmetic is broken into fragments. This arrangement makes it difDifficulty of ficult to realize that these two systems of num tracing the connection of bers differ from each other in no essential parthe fractions. ticular; that they are both formed from the unit one by the same process, with only a slight modification of the scale of division. ARITHMETICAL LANGUAGE. § 192. We have seen that the arithmetical al- Arithmetical phabet contains ten characters.* From these elements the entire language is formed; and we now propose to show in how simple a manner. alphabet. characters. First ten combina tions. The names of the ten characters are the first Names of the ten words of the language. If the unit 1 be added to each of the numbers from 1 to 10 inclusive, we find the first ten combinations in arithmetic. If 2 be added, in like manner, we have the second ten combinations; adding Second ten, 3, gives us the third ten combinations; and so on, until we have reached one hundred combinations (page 123). and so on for others. ing one addi Now, as we progressed, each set of combina- Each set givtions introduced one additional word, and the tional word. results of all the combinations are expressed by the words from two to twenty inclusive. § 193. These one hundred elementary combinations, are all that need be committed to memory; for, every other is deduced from them. They are, in fact, but different spellings of the first nineteen words which follow one. If we extend the words to one hundred, and recollect that All that need be commit ted to me mory. at one hundred, we begin to repeat the numbers, Words to be we see that we have but one hundred words to remembered for addition. be remembered for addition; and of these, all Only ten above ten are derivative. To this number, words primitive. must of course be added the few words which express the sums of the hundreds, thousands, &c. Subtraction: § 194. In Subtraction, we also find one hundred elementary combinations; the results of which are to be read.* These results, and all Number of the numbers employed in obtaining them, are expressed by twenty words. words. Multiplica § 195. In Multiplication (the table being cartion: ried to twelve), we have one hundred and fortyfour elementary combinations, and fifty-nine Number of separate words (already known) to express the results of these combinations. words. Division: § 196. In Division, also, we have one hundred and forty-four elementary combinations, but words. use only twelve words to express their results. Number of Four hun dred and ighty-eight combina § 197. Thus, we have four hundred and eighelementary ty-eight elementary combinations. The results tions. of these combinations are expressed by one hunWords used: dred words; viz. nineteen in addition, ten in sub 19 in addi tion, 10 in subtrac tion, 59 in multiplication, traction, fifty-nine in multiplication, and twelve * Section 120. † Section 122. + Section 123. in division. Of the nineteen words which are 12 in division. employed to express the results of the combinations in addition, eight are again used to express similar results in subtraction. Of the fifty-nine which express the results of the combinations in multiplication, sixteen had been used to express similar results in addition, and one in subtraction; and the entire twelve, which express the results of the combinations in division, had been used to express results of previous combinations. Hence, the results of all the elementary combinations, in the four ground rules, are expressed by sixty-three different words; and Sixty-three they are the only words employed to translate words in all. these results from the arithmetical into our com mon language. The language for fractional units is similar in every particular. By means of a language thus formed we deduce every principle in the science of numbers. different Language fractions. the same for arithmetic: § 198. Expressing these ideas and their combinations by figures, gives rise to the language Language of of arithmetic. By the aid of this language we not only unfold the principles of the science, Its value and but are enabled to apply these principles to every question of a practical nature, involving the use of figures. use. But few combinations which § 199. There is but one further idea to be : presented it is this, that there are very few change the combinations made among the figures, which of the figures. change, at all, their signification. signification Examples. First: Second: Selecting any two of the figures, as 3 and 5, for example, we see at once that there are but three ways of writing them, that will at all change their signification. First, write them by the side of each other Second, write them, the one over the other } 3 5, 5 3. , Third. Third, place a decimal point before .3, each .5. Now, each manner of writing gives a differ ent signification to both the figures. Use, how Learn the ever, has established that signification, and we language by know it, as soon as we have learned the lan use. words, and their uses. guage. We have thus explained what we mean by the arithmetical language. Its grammar em Its grammar: braces the names of its elementary signs, or Alphabet Alphabet,the formation and number of its words,—and the laws by which figures are connected for the purpose of expressing ideas. We feel that there is simplicity and beauty in this system, and hope it may be useful. |