Now the only use of the algebraic signs + and = is that they save time in writing and room in printing, and when persons become accustomed to their use they make plain a number of operations at a single glance, as will be shown hereafter. In the same way that the sign + means _added to_, the sign - means _less_ or subtracted from, thus: 1,872 - 468 = 1,404, which is the same as though it was printed as follows: 1,872 _less_ 468 _is equal to_ 1,404. The sign × means _multiplied by_, or is the sign of multiplication. Thus: 1,872 × 468 = 876,096; that is, 1,872 _multiplied by_ 468 _is equal to_ 876,096. The sign ÷ means divided by, thus: 1,872 ÷ 468 = 4. which means: 1,872 _divided by_ 468 _is equal to_ 4. The same thing is expressed by putting a line under the dividend and writing the divisor under the line, thus: 1,872 ----- = 4. 468 These signs are combined in various ways. Thus, supposing we wanted to add 1,872 to 468 and then divide the sum by 117, it would be necessary, in order to represent the arithmetical calculation, to do it as follows: 1872 468 ---- 117)2340(20 234 ---- 0 Algebraically it would be stated thus: 1872 + 468 ---------- = 20 117 If you wanted to add 124 to the quotient 20 above, the calculation would be as follows: 1872 468 ---- 117)2340( 20 234 124 ---- --- 0 144 This operation could be expressed by writing it as follows: 1872 + 468 ---------- + 124 = 144. 117 If we wanted to multiply the quotient 20 by 124 we would simply put the sign × instead of + before 124, thus: 1872 + 468 ---------- × 124 = 2480. 117 The sign of subtraction or division can be used in the same way. With these explanations it is believed that any one, with nothing more than an ordinary knowledge of the four elementary rules of arithmetic, can understand all the mathematics contained in the following pages. A little explanation may also be needed of the method of representing machinery and other structures by mechanical drawings. [Illustration: A B C D] If we want to represent the outside of any object, say an apple, we make a drawing of it as shown at _A_. Now if we want to show the inside of the apple, say the seeds and core, we can cut it in half and represent it as shown at _C_, which is then called a _section_ or _sectional view_ of the apple. If we represent it as it will appear if we are above it and looking down on it as shown at _B_, it is called a _top view_ or _plan_.