division draw a line parallel to the plain scale. Draw a line from division 10 on the perpendicular to division 9 on the plain scale; and parallel to this, draw lines from the other eight divisions on the first half inch on the plain scale.

The mode of using the diagonal scale will be best seen by the following examples :

The distances between the dots on the several lines beginning from the lowest are respectively 3.6, 2.91, 3.42, 1.53, 4.64, 3.85, 1.26, 4.07, 1.18, 5.99 half inches. From these examples, it appears that the first decimal place is shown on the plain scale, and the second place on the perpendicular, and that the whole decimal number is taken from the intersection of the lines drawn from these points of division, to whatever perpendicular on the plain scale shows the whole number before the given decimal. Thus, for the number 3.48: from the point marked, where the diagonal line from 4-tenths in the plain scale intersects the horizontal line from 8 hundredths in the perpendicular, to a similarly-marked point on the perpendicular from 3, the distance will be

3.48 of half an inch.

Fig. 4 would be the same for a diagonal scale of 200 feet to an inch, or of 300 yards to 1 inches, etc. The diagonal scale is serviceable in other than decimal divisions, as in the following case:-Make a diagonal scale of 7 feet to an inch, to show inches. Divide the inch in the plain scale into 7 equal parts; carry on 5 of these as in Fig. 3, and take 12 divisions on the perpendicular.

EXAMPLES.

1. Make a diagonal scale to show of an inch.

2. Make a diagonal scale of 1 inch to a foot, to show tenths of inches.

3. A Land-chain measure 66 feet. Make a diagonal scale of 2 chains to an inch, to show feet.

4. Make a diagonal scale of three yards to an inch, to show inches.

The following Questions are partly selected from examination papers, and are capable of being solved by the principles explained before; and, as they require in their solution the aid of every part of the work, they will furnish a test of the student's acquaintance with the subject.

MISCELLANEOUS QUESTIONS.

1. Show on a plain scale of inches, the following dimensions, 4.1, 37, 41, 11, 13, 1.9, 11⁄2 inches.

2. Show on a diagonal scale, 2.75, 3.04, 4.16, 9.98, 23, 75, 1.29 inches.

3. Cut off an arc of 25° from a circle of 1 inches radius, and divide it into 16 equal parts.

4. Make an angle of 1493° by the scale of chords.

5. Divide a line 3 inches long, in the proportion of the numbers 21, 31, 6.5, 3.75, 4.5.

6. Find an Harmonical mean between two lines 2 inches and 3 inches long respectively.

7. Find a fourth proportional to 3, 3.5, 4 inches.

8. Divide 3 inches into extreme and mean ratio.

9. On one side of a regular hexagon, describe a segment of a circle containing an angle of 41°. From the extremities of the other sides draw tangents to its arc.

10. Make a triangle A B C, having the angles ABC and A C B 35° and 634°, and B C 2 inches. On BC as base, describe a triangle outwardly, having its vertical angle 98, and on AC and AB, others with vertical angles 145° and 1161.° Describe a circle about either of these four triangles, and its circumference will pass through the vertices of all the other triangles. Show how this fact might be deduced from Euclid, Book III. Prop. 22.

11. By the aid of Prob. XXXIII, make an angle of 17210.

12. On a line 13 inches long, describe triangles whose vertical angles are 24°, 29°, 48° respectively.

13. From a circle of 11⁄2 inches radius, cut off a segment to contain 64°, and from this segment cut off another, to contain 96°.

14. What is the difference between contiguous and adjacent angles ?

15. Describe a regular pentagon of 13 inches side. On either side, within the figure, construct two isosceles triangles, having vertical angles 72° and 36°. On what particular points do their vertices fall?

16. Make a regular octagon on a line 13 inches long, and describe a circle about it.

17. In a circle, describe a regular polygon of 17 sides. 18. Can a circle be described about a rhombus, or rhomboid. See Euclid, Book III. Prop. 22.

19. Make an ellipse whose transverse diameter is 3 inches, and conjugate diameter 13 inches.

20. On a line 13 inches long, as transverse diameter, describe an oval.

21. Make an irregular pentagon of the following dimensions:-The sides AB, BC, CD, 2, 3, and 4 inches respectively, the angles A B C, BCD, 94° and 135°, and the other sides AE, ED, 4 and 5 inches respectively. On a side homologous to BC, 1 inch long, describe a similar figure.

22. Make a right-angled triangle equal to an equilateral triangle of 13 inches side.

23. Make a plain scale of § of an inch to a foot.

24. Make triangles equal to each of the figures in Ex. 21, and show by Prob. XVII. that the homologous sides of the figures are proportional to those of the triangles.

25. Make an isosceles triangle equal to a regular heptagon, having each side of an inch.

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26. Given an isosceles triangle, having the base 1 inches, and each of the angles at the base 472°, to make another isosceles triangle equal to it, with altitude 2 inches.

27. Describe a circle, with diameter 2 inches; apply successively five chords, measuring each 75 of an inch; make a chord to the remaining arc, and make a square equal to the rectilineal figure thus formed.

28. Describe a pentagon of 11⁄2 inches side; on one side, as base, describe an isosceles triangle within the figure, having its vertical angle 80°. Make another triangle equal to this, having its vertex on that of the pentagon.

29. Describe a segment on a line 24 inches long, whose angle is 72°, and make a triangle equal to it.

30. Make an octagon equal to a sector of a circle of 2 inches radius, and angle 594°.

31. Make a square equal to the sum of five squares, whose sides are respectively 1, 2, 2.5, 3, 3.5 inches.

32. Make a regular pentagon equal to the sum of the following figures:-a square of 1 inch side, a hexagon of inch side, and a circle of 1 inch radius.

33. Subtract a triangle, having its sides respectively 2, 3, 4 inches, from a circle whose diameter is 3 inches.

34. Make a diagonal scale of 3 yards to 75 of an inch, to measure inches.

35. Multiply a square of inch side by six.

36. Divide a square into two other squares which shall have the ratio of 5 to 7.

37. Make a plain scale of 495 feet to 3 inches.

38. Make a diagonal scale of 275 feet to 3 inches, to measure single feet. On this scale show 49 feet.

39. A cylinder of

a steam-engine is 7 feet 6 inches

in height, and is to be represented in a drawing of the Make a diagonal scale for it to

length of 5 inches. show fifths of inches.

40. Two drawings of the same map are made to the scales of 1 miles and 3 miles to an inch respectively. Construct the scales, and show what length a distance of 3 miles 110 yards will appear in each map.

THE END.

LONDON: KNIGHT AND SON, PRINTERS, CLERKENWELL CLOSE.