corresponding map distance by reducing the real distance to inches and multiplying by the R.F., or which is the same thing if the numerator of the R.F. is I divide the real distance, when reduced to inches, by the denominator of the R.F.

Suppose a straight line of 3 inches represents 20 feet and a scale is required to measure 50 feet: we take the representative fraction == 80. Divide 50 feet by 80 we obtain 7 inches, the length of the scale.

We could also have obtained the same result by remembering that 20 feet has to 50 feet the proportion of 2 to 5 and have found a fourth proportional (39) to the numbers 2, 5, 3.

If the map distance and R.F. are given to find the real distance divide the map distance by the R.F. As an example suppose the R.F. is and the map distance 5 inches. Divide 5 inches by or which is the same thing multiply by 60 we obtain 300 inches 25 feet.

=

The proposition upon which the diagonal scale depends is that if we divide two sides of a triangle into any number of equal parts by straight lines parallel to the base each of these parallels is greater in length than the one before it by the same amount, which is the length of the least parallel.

For example suppose the sides of the triangle AFG are

divided into three equal parts by DE and BC parallel to the base FG, then BC= of FG, DE=3 of FG. Similarly if AF and AG were divided into ten equal parts by straight lines parallel to the base FG these lines would be respectively 10. 1o, fo, 10, Yo, fo, fo, fo, o of FG.

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When two different scales are used for the same map, for instance an English scale of inches and a French scale of centimetres, they are said to be comparative. All we need remember is that comparative scales have the same representative fraction.

(109) To draw a plain scale.

In a certain map 2 miles are represented by 24 inches, draw a plain scale to measure up to 5 miles, shewing furlongs.

We can find the distance that represents 5 miles by the arithmetical rules just given or by finding a fourth proportional to 2, 21, 5 (39).

By arithmetic the working is as follows: the R.F. is 2 inches divided by 2 miles reduced to inches

Divide this straight line into 5 equal parts, each of these parts represents one mile. Take the last division on the left, and on the right of it put THE ZERO POINT. Divide this last division into eight equal parts to shew furlongs.

In drawing the scale first make the lower line moderately thick, and of the length required, then make two perpendiculars at the ends about the same length as shewn in the example. Next draw a thin line parallel to the first and terminated by the perpendiculars, and bisecting them, or nearly so.

The perpendiculars drawn at the points of division marking miles should be drawn equal to those at the ends of the line, but those marking furlongs only to meet the thin line.

In figuring the scale observe that the miles read from the zero point to the right, and the furlongs, from the zero point to the left.

To measure a distance from the scale, for example 3 miles 5 furlongs, we measure from the 3 of the miles to the 5 of the furlongs, as marked by asterisks.

(110) To draw a diagonal scale.

In a certain map 2 miles are represented by 23 miles, draw a diagonal scale to measure miles, furlongs, and chains, any distance up to 5 miles.

As in the last example we find the length required to represent 5 miles is 5 inches. This distance is, as before, divided into 5 equal parts, and the last division on the left-hand side into 8 equal parts to shew furlongs.

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67

The zero point occupies the same position and the figuring of the scale as far as regards the miles and furlongs is also the same. In drawing the scale, however, the perpendiculars drawn at the end of the first line are now an inch or more in length.

A number of parallel straight lines are now drawn to the first line, in this case ten, terminated by the perpendiculars at the end of the line, all at the same distance from one another. This can be done by marquoise scales, or by marking off equal distances on the perpendiculars.

Now draw perpendiculars at each of the divisions marking miles, by making parallels to those at the ends. Divide the last division on the highest parallel into eight equal parts corresponding to the eight divisions on the base line (or first line drawn). Join the zero point to the first of these divisions, the first on the base line to the second on the highest line, and so on, to obtain the slanting lines. The parallels are numbered I to 10 from the lowest. It is sufficient to put the figures on alternate lines. As explained on page 65 we thus obtain 10, fo, io, &c. of 1 furlong, that is 1, 2, 3, 4...chains.

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109

As an example of measuring any distance, take 2 miles 3 furlongs 6 chains. Starting from the base line division for 3 furlongs follow up the slanting line to where it crosses the parallel for 6 chains, this gives the first point. The second point is where this parallel cuts the perpendicular at the 2 miles division.

To understand how it is that this length does measure the distance given we observe that the part on the parallel between the small dots is of the part representing one furlong on the highest parallel, that is 6 chains, it can easily be seen that the parts on the right and left of the dots represent respectively 2 miles and 3 furlongs, so that the total is 2 miles, 3 furlongs,

6 chains.

A diagonal scale can be used in the same way to shew yards, feet and inches.

Divide the line given to shew the required number of yards. Put the zero point at the right-hand side of the last division on the left. Divide this division into three equal parts to shew feet.

Let the figuring always run to the right and left of the zero point, always reading from the zero point.

Complete the drawing as before, but in this case draw 12 parallels to the base (not 10) to shew inches.

To measure then 3 yards, 2 feet, 7 inches.

Find the point indicating 2 feet, follow the slanting line from it to the seventh parallel, (for the seven inches,) and follow along this parallel to where it crosses the perpendicular for 3 yards, this gives the second point.

The diagonal scale on the protractor giving inches, or half inches to hundredth parts is that most used for drawing where inches and decimals of inches are required.

Take the diagonal scale representing inches and hundredths of inches. The base line is divided into a certain number of inches, the last division on the left into ten equal parts shewing tenths of inches, the zero point is on the right of this division, and there are ten parallels dividing each tenth into ten parts, that is into hundredths.

To measure 3.42 inches. Find the point indicating 4 or of an inch, follow along the slanting line drawn from this point to where it meets the second parallel to shew 1, the second point on which to place the compass is where this second parallel meets the perpendicular through 3 inches.