It occurs sometimes that a student familiar only with computation by numerals is needlessly puzzled, in approaching the subject of Algebra, to comprehend how it is possible to multiply letters together, or to divide them. To remove this difficulty, it may be sufficient for them to learn that their perplexity arises from a misunderstanding in supposing the letters themselves are ever multiplied or divided. It is true that in treatises on the subject it is usual to speak as though these operations were actually performed upon the letters. It is always understood, however, that it is not the letters, but the quantities represented by the letters, which are to be multiplied or divided.
For example, in Art. 361 it is shown, in comparing similar sides of homologous triangles, that the bases of the two triangles are to each other as the corresponding sides, or, referring to Fig. 269, we have CE: A E:: D E: B E. Now, let the two bases C E and A E be represented respectively by a and b, and the two corresponding sides D E and B E by c and d respectively; or, for -
CE: AE:: DE: BE, put -
a: b:: c: d;
and, by Art. 373, we have -
b x c = a x d, which may be written -
bc = ad;
for x, the sign for multiplication, is not needed between letters, as it is between numeral factors. The operation of multiplication is always understood when letters are placed side by side.
Now, here we have an equation in which, as usually read, we have the product of b and c equal to the product of a and d. But the meaning is that the product of the quantities represented by b and c is equal to the product of the quantities represented by a and d, and that this equation is intended to represent the relation subsisting between the four proportionals, C E, A E, D E, and BE, of Fig. 269. In order to secure greater conciseness and clearness, the four small letters are substituted for the four pair of capital letters, which are used to indicate the lines of the figures referred to.
397. - Example: Application. - It was shown in the last article that the four letters a, b, c, and d represent the corresponding sides of the two triangles of Fig. .269, and that -
b c = a d.
Now, let each member of this equation be divided by a, then (Art. 371) -
bc/a = d If now the dimensions of the three sides represented by a, b, and c are known, and it is required to ascertain from these the length of the side represented by d, let the three given dimensions be severally substituted for the letters representing them. For example, let a = 40 feet; b = 52 feet, and c = 45 feet; then -
d = bc/a = 52x45/40 = 2340/40 = 58.5 feet.
The quantities being here substituted for the letters; we have but to perform the arithmetical processes indicated to obtain the arithmetical value of d. From this example it is seen that before any practical use can be made of an algebraical formula in computing dimensions, it is requisite to substitute numerals for the letters and actually perform arithmetically such operations as are only indicated by the letters.