FORMULAS.

The term formula, as used in mathematics and in techni cal books, may be defined as a rule in which symbols are used instead of words; in fact, a formula may be regarded as a shorthand method of expressing a rule.

Most people having no knowledge of algebra regard formulas with distrust; they think that a person must be a good algebraic scholar in order to be able to use formulas. This idea, however, is erroneous. As a rule, no knowledge of any branch of mathematics except arithmetic is required to enable one to use a formula. Any formula can be expressed in words, and when so expressed it becomes a rule.

Formulas are much more convenient than rules; they show at a glance all the operations that are to be performed; they do not require to be read three or four times, as is the case with most rules, to enable one to understand their meaning; they take up much less space, both in the printed book and in one's note book, than rules; in short, whenever a rule can be expressed as a formula, the formula is to be preferred. In the following pages we purpose to show the reader how to use such formulas as he is likely to encounter in "pocketbooks," or other works of like nature.

The signs used in formulas are the ordinary signs indicative of operations and the signs of aggregation. All these signs are used in arithmetic, but, to refresh the reader's memory, we will explain their nature and uses before proceeding further.

The signs indicative of operations are six in number, viz.: +, -, X, ÷, 1, V.

The sign (+) indicates addition, and is called plus; when placed between two quantities, it indicates that the two quantities are to be added. Thus, in the expression 25+ 17, the sign (+) shows that 17 is to be added to 25.

The sign (-) indicates subtraction, and is called minus; when placed between two quantities, it indicates that the

quantity on the right is to be subtracted from that on the left. Thus, in the expression 25-17, the sign (—) shows that 17 is to be subtracted from 25.

The sign (X) indicates multiplication, and is read times, or multiplied by; when placed between two quantities, it indicates that the quantity on the left is to be multiplied by that on the right. Thus, in the expression 25 X 17, the sign (X) shows that 25 is to be multiplied by 17.

The sign (÷) indicates division, and is read divided by; when placed between two quantities, it indicates that the quantity on the left is to be divided by that on the right. Thus, in the expression 25 ÷ 17, the sign (÷) shows that 25 is to be divided by 17.

Division is also indicated by placing a straight line between the two quantities. Thus, 25 | 17, 25/17, and 1 all indicate that 25 is to be divided by 17. When both quantities are placed on the same horizontal line, the straight line indicates that the quantity on the left is to be divided by that on the right. When one quantity is below the other, the straight line between indicates that the quantity above the line is to be divided by the one below it.

The sign (V) indicates that some root of the quantity to the right is to be taken; it is called the radical sign. To indicate what root is to be taken, a small figure, called the index, is placed within the sign, this being always omitted when the square root is to be indicated. Thus, 25 indicates that the square root of 25 is to be taken; 25 indicates that the cube root of 25 is to be taken, etc.

NOTE. As the term "quantity" is a very convenient one to use, we will define it. In mathematics the word quantity is applied to anything that it is desired to subject to the ordinary operations of addition, subtraction, multiplication, etc., when we do not wish to be more specific and state exactly what the thing is. Thus, we can say "two or more numbers," or "two or more quantities." The word quantity is more general in its meaning than the word number.

The signs of aggregation are four in number, viz.: -, (), [], and {}, respectively calle the vinculum, the parenthesis, the brackets, and the brace; they are used when it is desired to indicate that all the quantities included by them

are to be subjected to the same operation. Thus, if we desire to indicate that the sum of 5 and 8 is to be multiplied by 7, and we do not wish to actually add 5 and 8 before indicating the multiplication, we may employ any one of the four signs of aggregation as here shown: 5+8X7, (5+8) X7, [5+8] X7, {5+87. The vinculum is placed above the quantities which are to be treated as one quantity and subjected to the same operations.

While any one of the four signs may be used as shown above, custom has restricted their use somewhat. The vincu lum is rarely used except in connection with the radical sign. Thus, instead of writing (5+8), V3 [5 +8], or † { 5 +8} for the cube root of 5 plus 8, all of which would be correct, the vinculum is nearly always used, 5+8.

In cases where but one sign of aggregation is needed (except, of course, when a root is to be indicated), the parenthesis is always used. Hence, (5+8) X7 would be the usual way of expressing the product of 5 plus 8 and 7.

If two signs of aggregation are needed, the brackets and parenthesis are used, so as to avoid having a parenthesis within a parenthesis, the brackets being placed outside. For example, [(205)÷3] X 9 means that the difference between 20 and 5 is to be divided by 3, and this result multiplied by 9. If three signs of aggregation are required, the brace, brackets, and parenthesis are used, the brace being placed outside, the brackets next, and the parenthesis inside. For example, {[(205) ÷ 3] × 9—21 } ÷ 8 means that the quotient obtained by dividing the difference between 20 and 5 by 3 is to be multiplied by 9; and that 21 is to be subtracted from the product thus obtained, and the result divided by 8.

Should it be necessary to use all four signs of aggregation, the brace would be put outside, the brackets next, the parenthesis next, and the vinculum inside. For example, {[(20-53) X9-21]÷8×12. The reason for using the brace in this last instance will be explained, as it is not generally understood.

When several quantities are connected by the various signs indicating addition, subtraction, multiplication, and division, the operation indicated by the sign of multiplication

must always be performed first. Thus, 2 + 3 X 4 equals 14, 3 being multiplied by 4 before adding to 2. Similarly, 10 ÷ 2 × 5 equals 1, since 2 × 5 equals 10, and 10 ÷ 10 equals 1. Hence, in the above case, if the brace were omitted, the result would be; whereas, by inserting the brace, the result is 36.

Following the sign of multiplication comes the sign of division in its order of importance. For example, 5-9÷3 equals 2, 9 being divided by 3 before subtracting from 5. The signs of addition and subtraction are of equal value; that is, if several quantities are connected by plus and minus signs, the indicated operations may be performed in the order in which the quantities are placed.

There is one other sign used, which is neither a sign of aggregation nor a sign indicative of an operation to be performed; it is (=), and is called the sign of equality; it means that all on one side of it is exactly equal to all on the other side. For example, 2 = 2, 5-32, 5X (14 — 9) : = 25.

It is customary to omit the sign of multiplication between two or more quantities when they are to be multiplied together, or between a number and a letter representing a quantity, it being always understood that when two letters are adjacent with no sign between them, the quantities represented by these letters are to be multiplied.

The sign of multiplication, evidently, cannot be omitted between two or more numbers, as it would then be impossible to distinguish the numbers. A near approach to this, however, may be attained by placing a dot between the numbers that are to be multiplied together, and this is frequently done in works on mathematics when it is desired to economize space. In such cases it is usual to put the dot higher than the position occupied by the decimal point. Thus, 2:3 means the same as 2X3; 542 749 1,006 indicates that the numbers 542, 749, and 1,006 are to be multiplied together.

It is also customary to omit the sign of multiplication in expressions similar to the following: a b + c, 3 × (b + c), (b+c) xa, etc., writing them a b+c, 3 (b + c), (b + c) a, etc. The sign is not omitted when several quantities are included by a vinculum, and it is desired to indicate that the quantities