ARITHMETIC, GEOMETRY, DRAWING, ETC

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**1.** We often hear the theory advanced that in this country, at the present day, it is not necessary to have a knowledge of arithmetic, geometry, etc., because our interchangeable system of manufacturing watches, makes all knowledge in these lines superfluous, and that without any knowledge of arithmetic or geometry a man may become a thorough master of watch-making. This is a mistake that too many of our young men make. The fact that the leading watch factories of the United States have adopted the interchangeable system, of course lessens the number of parts which the repairer will have to make and fit, but it by no means alters the situation as regards the repairing of foreign-made watches, nor even the changing of American watches from key to stem-winders. Without a thorough knowledge of arithmetic and, at least, an insight into the principles of geometry, no young man can hope to become a first-class watch-maker in the true sense of the word. Without these accomplishments he will be deprived of the pleasure of reading understandingly the best literature of the day, the works of those who are best fitted to impart knowledge to the members of the trade.

With a knowledge of geometry he will be able to comprehend the works of the best authors, to ascertain the dimensions of solid bodies, and be in a position to apply the rules that form the basis of linear drawing. Every watch-maker, worthy of the name, should be able to make and understand the drawing of any machine, or of any horological instrument. Many inventors, and even ordinary workmen, would avoid a large amount of hard work, often useless, and occupying much time, if, instead of at once putting an idea into practice with brass and steel, they were able as a preliminary, to make for themselves a correct design drawn to scale.

**2.** It is taken for granted that the reader is familiar with the rules of arithmetic at least, and we will touch upon some points in algebra and geometry that it will be well to mention. Should the reader have no knowledge of arithmetic, algebra and geometry, we would advise him to take up these studies during his leisure hours, using some of the standard text books for that purpose. (*) Besides possessing a knowledge of prime numbers, numbers which have no divisors but unity and themselves, the watch-maker should be able to determine the greatest common measure of several numbers, a rule which is of great importance in calculating a train of wheels that is complicated.

We shall confine our attention to the methods of extracting square roots and proportions, the rules for which may have been forgotten, owing to their being less frequently employed than the more common rules of arithmetic; they are of frequent use in horology.

**3.** The sign of addition is an erect cross, +, called *plus*, and when placed between two quantities it indicates that the second is to be added to the first. Thus, 5 + 3 equals 8.

The sign of subtraction is a short horizontal line, -, called *minus*, and when placed between two quantities it indicates that the second is to be subtracted from the first. Thus, 8 - 3 equals 5.

The double sign, ±, is sometimes written before a quantity to indicate that in certain cases it is to be added and in others it is to be subtracted. Thus 5 ± 3 is read 5 *plus or minus* 3.

The sign of multiplication, ×, when placed between two quantities indicates that the first is to be multiplied by the second. Thus, 3 × 5 equals 15.

The sign of division is a short horizontal line with a point above and one below, ÷, and when placed between two numbers or quantities it indicates that the first is to be divided by the second. Thus, 6 ÷ 2 equals 3.

A parenthesis, (), denotes that the several numbers or quantities included within it are to be considered together, and subjected to the same operation. Thus, (10 + 4) × 3 indicates that both 10 and 4, or their sum, are to be multiplied by 3.

A horizontal vinculum, ________, placed over the numbers or quantities, is frequently used instead of the parenthesis. Thus, is equivalent to (2+4+6)×7.

Division is more usually indicated by a line between the two figures, the dividend being written above and the divisor below the line. Thus, 16 / 8 indicates division, the same as 16 ÷ 8.

Algebra is that branch of mathematics in which the operations are indicated by signs or symbols, and the quantities are represented by letters.

The sign of ratio consists of two points like the *colon*, :, aplaced between the quantities compared. Thus, the ratio of *a* to *b* is written *a:b*.

The sign of proportion consists of a combination of the signs ratio. Thus **: :: :**. The first two and the last two dots are read *is to*, while the four in the midle are read *as*.. Thus, *a, b, c,* and *d*, are four quantities which are proportional to each other, we say *a* is to *b* as *c* is to *d*, and is expressed by writing them thus;

**4.** The power of a number is the product formed by successive multiplication of the same number by itself. Thus,

- 2×2=4, the second power or square of 2.
- 2×2×2=8. the third power or cube of 2.
- 2×2×2×2=16, the forth power of 2.
- Etc.

An exponent is a number written above a quantity, at the right hand, to indicate how many times the quantity is to be taken as a factor, as 6³=6×6×6.

The root of a quantity is a factor which, multiplied by itself a certain number of times, will produce the given quantity. Thus, in the above examples 2 is the square root of 4 and the cube root of 8.

The radical sign, √, indicates that the root of the quantity placed under it is to be taken, and the index of the bend. If a square root the index figure is usually omitted.

**5. Extracting the square root of whole numbers. (**)**

**I.** Point the given number off into periods of two figures each, counting from the units place to the left. For example, we wish to find the square root of 399427, we point it off thus: 39,94,24.

**II.** Find the greatest perfect square in the left hand period, and write its root for the first figure in the required root; subtract the square of figure from the first period, and to the remainder bring down the next period for a dividend.

**III.** Double the root already found, and write the result on the left for a divisor; find haw many times this divisor is contained in the dividend, exclusive of the right-hand figure, and place the result in the root and at the right of the divisor.

**IV.** Multiply the divisor thus completed by the last figure of the root; subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. Thus:

**V.** Double the right-hand figure of the last complete divisor for a new divisor, and continue the operation as before. Thus:

**6.** It is often convenient to express the relations of qualities in the form of a proportion and from the proportion derive an equation.

Ratio is the quotient of one number divided by another. Thus the ratio of 30 to 6 is 30/6.

**7.** Proportion is the equality of ratios; Thus if 30/6=5 and 40/8=5 then we may state that 30/6=40/8 or the proportionality is usually expressed (**3**) thus: 30:6::40:8 and this constitutes what is called a geometrical proportion, and 30 and 8 are called the extremes and 6 and 40 the means.

**8.** The product of the extremes is always equal to the product of the means. Thus: 30×8=6×40=240. Hence it follows that if we only know three terms we can always determine the fourth, or unknown term, which is usually represented by the letter *x*. Thus in the proportion 12:3::16:x we find the product of the means, or 3×16=48; this product divided by 12, the known extreme, gives us the value of x, or the unknown extreme, as equalling 4.

**9.** If we know the two extremes and only one of the means the same rule is applied. Thus in the proportion 20:5::x:25 we have: 20×25=500. 500/5=100, the value of x.

(*) Loomis' Tretise on Arithmetic

Loomis' Tretise on Algebra

Loomis' Elements of Geometry

Robinson's Algebra and Geometry

(**) Adapted from Robinson's Algebra

Submitted by: Samuel Kirk (##)