between any two points is two-thirds the horizontal distance; the denominator of the fraction, in all cases, representing the number of parts in horizontal projection, and the numerator the corresponding number of parts in vertical distance.

When the position of a line is designated in this way, it is said to be a line whose inclination or declivity is one-sixth, two-thirds, ten on one, &c., or simply, a line of one-sixth, &c.

6. Having the declivity of a line, the difference of reference of any two of its points, the projections of which are given, will be found by multiplying the horizontal distance between them by the fraction which expresses this declivity; in like manner the horizontal distance of any two points will be obtained by dividing the difference of their references by this fraction.

To obtain therefore the reference of a point of a line, having its projection, the horizontal distance between it and that of some other known point of the line must be determined from the scale of the drawing by which the horizontal distances are measured; this distance expressed in numbers, being multiplied by the fraction which expresses the declivity of the line, will give the difference of reference of the two points; the required reference of the point will be found by subtracting this product from the reference of the known point, if it is higher than the one sought, or adding if it is lower. Thus let (25.15) be the reference of a known point higher than the one sought; the horizontal distance between the points being 35.75 feet, and the inclination of the line 7%; then 35.75 x 3.575 will be the difference of reference of the points, and 25.15 - 3.575 21.575, the required reference. The converse of this shows that the horizontal distance between two points on this line whose difference of reference is 3.575 will be 3.575÷35.75 feet.

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7. When the projection of a line is divided into equal parts, each of which corresponds to a unit in vertical distance, and the references of the points of division are written, it is termed the scale of declivity of the line. In constructing the scale of declivity of a line, the entire references are alone put down; one of the divisions of the equal parts being subdivided into tenths, or hundredths if necessary, so as to give the fractional parts of the references corresponding to any fractional part of an entire division.

3. The true length of any portion of an oblique line be tween two given points is evidently the hypothenuse of a right angle triangle of which the other two sides are the dif ference of reference of the points, and their horizontal dis

tance.

9. Plane. The position of a plane oblique to the plane of reference may be determined either by the projections and references of three of its points; by the projections and declivity of two lines in it oblique to the plane of reference; or by the projection of two or more horizontal lines of the plane with their references.

The more usual method of representing a plane is by the projections on the plane of reference of the horizontal lines determined by intersecting it by equidistant horizontal planes. These projections are termed horizontals of the plane, those usually being taken the references of which are entire numbers.

10. If in a given plane a line be drawn perpendicular to any horizontal line in it, the projection of this line on the plane of reference will be also perpendicular to the projections of the horizontals. The angle of this line with the plane of reference is evidently the same as that of the given plane with it, and is greater than the angle between any other line drawn in the plane and the plane of reference. This line is, on this account, termed the line of greatest declivity of the plane.

11. If the scale of declivity of the line of greatest declivity be constructed, it will alone serve to fix the position of the plane to which it belongs, and to determine the reference of any point of the plane of which the projection is given. For since the horizontals are perpendicular to the scale of declivity, the point where the horizontal drawn through the given projection of a point in the plane cuts this line will determine upon the scale the reference of the horizontal, and therefore that of the point.

12. The inclination or declivity of a plane with the plane of reference may be expressed in the same way as the inclination of its line of greatest declivity. Thus a plane of onefourth; a plane of twenty on one; a plane of two-thirds, express that the natural tangents of the angle between the planes and the plane of reference are respectively represented by the fractions 1, 2, and 3.

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13. The horizontal distance between any two horizontal lines in a plane, the angle of which is given, can be found in the same way as the horizontal distance between two points of a line, the inclination of which is given, Art. 7, by dividing the difference of the reference of the two horizontal lines by the fraction representing the declivity of the plane; in like manner the difference of references of any two horizontal lines will be obtained by multiplying their horizontal distance by the same fraction.

14. To distinguish the scale of declivity, Pl. 1, Fig. 2,

from any other line of a plane, it is always represented by two fine parallel lines, drawn near each other, and crossed at the points of division, where the references are written, by short lines which are portions of the corresponding horizontals.

With the foregoing elements the usual problems of the right line and plane can be readily solved.

15. Problems of the Right Line and Plane. Prob. 1, Pl. 1, Fig. 3. Having the projections and references of two lines that intersect, to find the angle between them.

Let ab be the projection of one of the lines, the references of two of its points (10.30) and (4.90) being given; cd the projection of the other line, (10.30), and (5.0) being the references of two of its points; (10.30) being the point of intersection of the two lines.

Find on each of the lines, Art. 7, a point having the same reference (7.0). The line joining these two points will be horizontal, and projected into its true length; taking this line as the base of a triangle of which the other two sides are respectively the true lengths of the portions of the two given lines projected between (10.30) and (7.0), Art. 7, the angle at the vertex will be the one required.

16. Prob. 2, Fig. 4. Through a point to draw a line parallel to a given line.

Let c (7.50) be the projection of the point; ab that of the given line of which the two points (7.0) and (9.0) are

known.

Through c drawing ed parallel to ab, this will be the projection of the required line; and as its declivity is the same as that of the given line, it will be only necessary to set off from c towards d, the same distance as between (7.0) and (9.0), to obtain a point (9.50) as far above (7.50) as (9.0) is above (7.0).

17. Prob. 3, Fig. 5. Through a point in a plane to draw a line in the plane with a given inclination.

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Let cd be the scale of declivity of the given, plane, and a (5.50) the given point; and suppose, for example, that the declivity of the plane is and that of the required line is 7. Draw the horizontal of the plane (5.50) which passes through the point, and any other horizontal, as (7.0).ˆ_ The projection of the required line will pass through a, and the portion of it between the two horizontals will be equal, Art. 6, to the difference of their references, or 1.5 ft. divided by the fraction which represents the inclination of the required line. Describing, therefore, from a, an arc, with this distance ac or 1.515 ft. as a radius, and joining the

point b, where it cuts the horizontal (7.0), with a, this will be the projection of the required line.

18. Prob. 4, Pl. 1, Fig. 6. Having three points of a plane, to construct its horizontals and scale of declivity.

Let a (12.0), b (15.25), and c (15.50), be the projections of the three points. Join a with the other two, and construct the scales of declivity of the lines of junction, Art. 6. The lines joining the same references on these two scales will be horizontals of the required plane. Its scale of declivity is constructed by drawing two parallel lines perpendicular to the horizontals, and writing the references of the points where they intersect the horizontals.

19. Prob. 5, Pl. 1, Fig. 7. To find the horizontals of a plane passed through a given line and parallel to another line.

Let ab and cd be the projections of the two lines. From a point (10.0) on cd draw a line df, Prob. 2, parallel to ab; and by Prob. 4 find the horizontals of the plane of df and cd; these will be the required horizontals.

20. Prob. 6, Pl. 1, Fig. 8. To find the horizontals of a plane the declivity of which is given, and which passes. through a given line.

Let bd be the scale of declivity of the given line, and suppose, for example, the declivity of the line to be and that of the required plane to be 1.

b

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Since the horizontals of the plane must pass through the points of the line having the like references, and as the distance in projection between any two of them, Art. 13, will be equal to the difference of their references divided by the fraction giving the declivity of the plane, it follows that to find the one drawn through 6 (14.0), for example, it will be simply necessary to describe from any other point, as a (12.0), an arc of a circle, with a radius of 12 ft., equal to the quotient just mentioned, and to draw a tangent to this are from b. If any other horizontal, as (16.0), is required, which would not intersect the projection of the given line within the limits of the drawing; any two points, as (12.0) and (14.0), for example, may be taken as centres, and two ares be described from them, with radii of 12 and 24 ft., calculated as above, and a line be drawn tangent to the arc; this tangent will be the required horizontal.

21. Prob. 7, Pl. 1, Fig. 9. Having either the horizontals or the scales of declivity of two planes, to find their intersec

tion.

Join the points ab where any two horizontals, as (12.0) and (14.0), in one plane intersect the corresponding horizon

tals of the other, and the line so drawn will be the projection of the required intersection.

22. When the horizontals of the two planes are parallel, or when they are so nearly parallel that their points of intersection cannot be accurately found, the following method may be taken: Draw any two parallel lines as cd, c'd', Pl. 1, Fig. 10; these may be considered as the horizontals of an arbitrary plane, and having the same references, (12.0) and (14.0), as the two corresponding horizontals in each of the given planes. The intersections of the horizontals of the arbitrary plane with those of the given planes will determine two lines, mn, m'n', which, being the projections of the intersections of the given planes with the arbitrary plane, will, by their intersection o, determine the projection of a point common to the three planes, and therefore a point of the projection of the intersection of the two given planes. Assuming any other two parallels ab, a'b', as the horizontals of another arbitrary plane; finding in like manner the point o' and joining o and o' by a line, this will be the required projection.

"When the horizontals of the two planes are parallel, one point, as o, will be sufficient to determine the required projection, as it will be parallel to the horizontals.

23. Prob. 8, Pl. 1, Fig. 11. To find where a given line pierces a given plane.

Through the projections of any two points of the given line, as m', n', having the same references, (12.0), (14.0), as two horizontals of the given plane, draw two parallel lines, ab, a'b', which may be taken as the horizontals of an arbitrary plane. The projection of the line of intersection, mn, of this plane with the given plane being determined by Prob. 7, the point o where it intersects the projection of the line m'n' will be the projection of the required point, the refer ence of which can be found from the scale of the plane.

24. Prob. 9, Pl. 1, Fig. 12. To draw from a given point a perpendicular to a given plane, and find its length. Let a (12.0) be the projection of the given point; and let the given plane be represented by its scale of declivity.

The projection of the required perpendicular will pass through a, and be parallel to the scale of declivity of the given plane. The angle which it makes with the plane of reference is the complement of that between this plane and the given plane; its tangent therefore will be the reciprocal of the tangent of that of the given plane.

Drawing therefore through a the line ac parallel to bd, and constructing its scale of declivity, Art. 7, this will be