Part 11

In fig. 305 we have a rough sectional diagram of the eye and an object in front of it. This object, an arrow, is seen by means of the visual rays proceeding from it, the principal two of which are shown. The visual ray from _A_ passes through the pupil and is received upon the retina at _a_. In the same way the visual ray from _B_ passes through the pupil and is received upon the retina at _b_. It will thus be seen that the impressions or images received upon the retina are inverted; but, by long reason and experience, the mind has acquired the habit of determining the real positions of objects, and does not, though the image is so received, imagine them to be upside down.

It will also be observed, in the same way, that that portion of an object which is upon the right will be pictured upon the retina upon the left, and _vice versa_, but the mind, for the reasons before stated, never imagines the object to be reversed. This fact is another proof that, as mentioned at the commencement of our study, to see accurately is a matter of education and practice.

And first of Optics; it was asserted, page 252, that we see by the agency of light which passes from objects to our eyes in straight lines which are called Visual Rays.

We see by the agency of light, as all objects, except such as may be styled self-luminous, when placed in a dark chamber are not perceivable by us, except by touch, smell or hearing; we cannot _see_ them; they are invisible. But when, by removing a shutter or igniting a flame, we introduce something to the chamber which was not present when the chamber was dark, we become at once conscious of the appearance of the object, we perceive it by the sense of sight.

This something which must always be present to enable us to see, is called Light; all objects are made visible to the sense of seeing by its agency.

Without light, natural or artificial, it would be impossible to distinguish one object from another.

That the Visual Rays pass from objects in straight lines to the eye may be proved by the following experiment (see fig. 306):--Pierce two screens with a large pin, and place them so that the holes are in a straight line with a flame, as the light of a candle or lamp. On fixing the eye to one of these holes we are able to see the flame; but if we slightly move the flame, one of the screens, or the eye, the flame is no longer visible. To be visible, the flame, the holes in the screens, and the eye must all be in the same straight line. See fig. 306.

In fig. 299 the picture plane is represented by the rectangle _W X Y Z_. Although the picture plane is here shown as a rectangle, it may be of any shape or of any size.

The observer is at _S_, looking through the picture plane at the cross _R C O H_. The observer is standing upon a horizontal surface, which is called the ground plane. If we are in a room, the window may be called a picture plane and the floor a ground plane.

The picture plane rests, as it were, upon the ground plane, in a line which passes from _Y_ to _Z_. The two planes meet or intersect in this line, which is called the ground line. The ground line is sometimes called the picture line, or the measuring line.

The visual rays, by means of which the observer sees the cross, will, in their course from it to the eye, pass through the picture plane. These visual rays will intersect the picture plane in a number of points, and if we mark the true positions of these points the result will be a perspective image of the cross.

The rays are shown passing from the cross to the eye of the observer, and meeting the picture plane in points _r_, _c_, _o_, _h_; _r_ being joined to _o_, and _c_ to _h_, we have the perspective image of the cross as it would appear to the observer at _S_. Of course an infinite number of rays proceed from the cross to the eye of the observer; but it is quite evident that we need only consider those proceeding from the extremities of the object.

Scale or Approximate Perspective.

Real, or true perspective, represents the object exactly as it is seen in nature, where the parts that are far away from the eye of the observer appear smaller than those nearby. Occasions arise, however, in practical life, with its numerous phases of industrial requirements, where the convenience of showing the complete form of the object in a single view might preferably be coupled with the convenience of scale dimensions.

This has led to a modified perspective, that sacrifices some of the accuracy in the appearance of the object to gain the advantage of scale dimensions; this form of perspective may be distinguished by the name--_approximate or scale perspective_--which does not represent the object exactly as seen in nature, but where those parts that are afar off are shown of the same size as those that are near by, and where the lines that run out into space are parallel to each other and do not converge into a vanishing point.

To represent an object in perspective, the horizon and the point of vision will have to appear in the drawing as the fundamental starting points.

Three dimensions are distinguished for the fixing of an object in space from a certain reference point. They are height, breadth and thickness, and are in their direction square to each other. The height is the fundamental direction, being derived from the direction of gravity, that invariably extends to the center of the earth.

All directions in the perspective determination of an object are parallel to these.

Vertical lines and planes point toward the center of the earth, while horizontal planes, including the directions of breadth and thickness, are square to the vertical direction. In this, the principal visual ray extends in the direction of thickness.

For a clear understanding of perspective, it must be firmly fixed in mind, that for each prominent point of the object behind the picture plane, a corresponding point lies in the picture plane, in that position where a straight line or ray of sight that is going from the eye to the point of the object, cuts through the picture plane.

Suppose we could replace these rays of sight by thin, visible threads of wire that would go through little holes in the picture plane, we could then walk around this bundle of rays, and by looking at it from three different directions, we would get three different views of it. We may look upon it from the top, from the side or from the end, where the bundle of rays all concentrate in the eye of the observer.

Figs. 307 and 308 show, in two cases, how these three views would appear. The end views are those where the perspective picture appears on the plane, while the top and side views only show where the rays intersect the picture plane. The top view shows how far, for example, point _A_ is distant from a vertical line _O Z_, while the side view shows how far point _A_ is below horizontal line _O X_, which is at the same height above the ground as the eye of the observer, _O_. Thus, all points of the cube can be located on the picture plane, and the outlines of the cube reproduced in perspective.

Modified arrangements are shown in figs. 309 and 310 for parallel and angular perspective.

The views are so arranged in relation to each other that the picture plane in the top view is parallel to the horizon and the ground-line, which latter is the intersection of the picture plane with the level ground of the end or perspective view. At the same time the eye of the observer is in one and the same vertical line for both views, two vanishing points may be found in the horizon outside of the principal visual ray. To find the position of these two vanishing points in the picture plane, the modified top view, fig. 310, is used.

As all lines that end in a vanishing point must be parallel in reality, this parallelism may be seen in the top view and lines through the eye of the observer, parallel to the directions of the main lines of the object, will cut the picture plane at the vanishing points.

Through these two vanishing points the directions of two sets of lines are found, the starting points of which are determined from the plane of measurement. The third set of lines, being vertical, also appears vertical and parallel in the picture.

The position of each vertical line is found in the top view, where the light rays from the observing eye to the ends of the vertical lines intersect with the picture plane. Projecting these points down upon the rays to the vanishing points produces the vertical lines in the picture.

For example, in fig. 311, the purpose of perspective is entirely defeated by placing the eye of the observer directly in front of the object and arriving at the view taken in mechanical drawing which needs supplementary views for complete comprehension of the form of the object.

In fig. 312 the eye of the observer is first placed directly opposite the object, then it sees the object to the left but a short distance away, while in the third figure the observer is farther away from the object. In each case the picture plane and plane of measurement is at the front face of the cube.

For such simple objects, it is not necessary to draw the top view at all. The only reminder of the top view is the eye or point of vision, the picture plane that falls together for the sake of convenience with the horizon of the end view and the ray that determines the measurement point _M_, which is, in this suppressed reproduction, absolutely necessary, in order to find the apparent position of the real corners behind the picture plane.

So far, only square or sharp-cornered objects have been represented in perspective.

It is evident, however, that round objects can also be shown in linear perspective, placing reference lines on the object and representing these as if they were real lines. A cylinder is thus shown in fig. 313 of which the end planes will appear very distinctly in sharp outlines.

Vertically, only the outlines of the cylinder, as contrasted against space, will appear as distinct outlines, while the reference lines will not appear and are therefore shown only as dotted lines.

Fig. 314 shows the approximate or scale perspective with all the axes drawn and the corresponding angles and scales marked. The outlines of the object running in these directions appear all parallel to the axes.

The approximate or scale perspective completely avoids all the difficulties of choosing a point of sight, of having several views, vanishing points and measurement points, and thus offers a representative view, with a great saving of time and labor. Particularly for mechanical purposes, where an artistic impression is not called for, it presents a distinct advantage over the true or real perspective.

Useful Tables for Draughtsmen.

TABLE OF DECIMAL EQUIVALENTS.

8ths, 16ths, 32ds and 64ths of an Inch.

8ths.

¹⁄₈ = .125 ¹⁄₄ = .250 ³⁄₈ = .375 ¹⁄₂ = .500 ⁵⁄₈ = .625 ³⁄₄ = .750 ⁷⁄₈ = .875

16ths.

¹⁄₁₆ = .0625 ³⁄₁₆ = .1875 ⁵⁄₁₆ = .3125 ⁷⁄₁₆ = .4375 ⁹⁄₁₆ = .5625 ¹¹⁄₁₆ = .6875 ¹³⁄₁₆ = .8125 ¹⁵⁄₁₆ = .9375

32nds.

¹⁄₃₂ = .03125 ³⁄₃₂ = .09375 ⁵⁄₃₂ = .15625 ⁷⁄₃₂ = .21875 ⁹⁄₃₂ = .28125 ¹¹⁄₃₂ = .34375 ¹³⁄₃₂ = .40625 ¹⁵⁄₃₂ = .46875 ¹⁷⁄₃₂ = .53125 ¹⁹⁄₃₂ = .59375 ²¹⁄₃₂ = .65625 ²³⁄₃₂ = .71875 ²⁵⁄₃₂ = .78125 ²⁷⁄₃₂ = .84375 ²⁹⁄₃₂ = .90625 ³¹⁄₃₂ = .96875

64ths.

¹⁄₆₄ = .015625 ³⁄₆₄ = .046875 ⁵⁄₆₄ = .078125 ⁷⁄₆₄ = .109375 ⁹⁄₆₄ = .140625 ¹¹⁄₆₄ = .171875 ¹³⁄₆₄ = .203125 ¹⁵⁄₆₄ = .234375 ¹⁷⁄₆₄ = .265625 ¹⁹⁄₆₄ = .296875 ²¹⁄₆₄ = .328125 ²³⁄₆₄ = .359375 ²⁵⁄₆₄ = .390625 ²⁷⁄₆₄ = .421875 ²⁹⁄₆₄ = .453125 ³¹⁄₆₄ = .484375 ³³⁄₆₄ = .515625 ³⁵⁄₆₄ = .546875 ³⁷⁄₆₄ = .578125 ³⁹⁄₆₄ = .609375 ⁴¹⁄₆₄ = .640625 ⁴³⁄₆₄ = .671875 ⁴⁵⁄₆₄ = .703125 ⁴⁷⁄₆₄ = .734375 ⁴⁹⁄₆₄ = .765625 ⁵¹⁄₆₄ = .796875 ⁵³⁄₆₄ = .828125 ⁵⁵⁄₆₄ = .859375 ⁵⁷⁄₆₄ = .890625 ⁵⁹⁄₆₄ = .921875 ⁶¹⁄₆₄ = .953125 ⁶³⁄₆₄ = .984375

TABLE OF DECIMAL EQUIVALENTS

Of Millimeters and Fractions of Millimeters.

_mm._ _Inches._

¹⁄₅₀ = .00079 ²⁄₅₀ = .00157 ³⁄₅₀ = .00236 ⁴⁄₅₀ = .00315 ⁵⁄₅₀ = .00394 ⁶⁄₅₀ = .00472 ⁷⁄₅₀ = .00551 ⁸⁄₅₀ = .00630 ⁹⁄₅₀ = .00709 ¹⁰⁄₅₀ = .00787 ¹¹⁄₅₀ = .00866 ¹²⁄₅₀ = .00945 ¹³⁄₅₀ = .01024 ¹⁴⁄₅₀ = .01102 ¹⁵⁄₅₀ = .01181 ¹⁶⁄₅₀ = .01260 ¹⁷⁄₅₀ = .01339 ¹⁸⁄₅₀ = .01417 ¹⁹⁄₅₀ = .01496 ²⁰⁄₅₀ = .01575 ²¹⁄₅₀ = .01654 ²²⁄₅₀ = .01732 ²³⁄₅₀ = .01811 ²⁴⁄₅₀ = .01890 ²⁵⁄₅₀ = .01969 ²⁶⁄₅₀ = .02047 ²⁷⁄₅₀ = .02126 ²⁸⁄₅₀ = .02205 ²⁹⁄₅₀ = .02283 ³⁰⁄₅₀ = .02362 ³¹⁄₅₀ = .02441 ³²⁄₅₀ = .02520 ³³⁄₅₀ = .02598 ³⁴⁄₅₀ = .02677 ³⁵⁄₅₀ = .02756 ³⁶⁄₅₀ = .02835 ³⁷⁄₅₀ = .02913 ³⁸⁄₅₀ = .02992 ³⁹⁄₅₀ = .03071 ⁴⁰⁄₅₀ = .03150 ⁴¹⁄₅₀ = .03228 ⁴²⁄₅₀ = .03307 ⁴³⁄₅₀ = .03386 ⁴⁴⁄₅₀ = .03465 ⁴⁵⁄₅₀ = .03543 ⁴⁶⁄₅₀ = .03622 ⁴⁷⁄₅₀ = .03701 ⁴⁸⁄₅₀ = .03780 ⁴⁹⁄₅₀ = .03858 1 = .03937 2 = .07874 3 = .11811 4 = .15748 5 = .19685 6 = .23622 7 = .27559 8 = .31496 9 = .35433 10 = .39370 11 = .43307 12 = .47244 13 = .51181 14 = .55118 15 = .59055 16 = .62992 17 = .66929 18 = .70866 19 = .74803 20 = .78740 21 = .82677 22 = .86614 23 = .90551 24 = .94488 25 = .98425 26 = 1.02362

10 mm. = 1 Centimeter = 0.3937 inches. 10 cm. = 1 Decimeter = 3.937 „ 10 dm. = 1 Meter = 39.37 „ 25.4 mm. = 1 English Inch.

RULES RELATIVE TO THE CIRCLE.

The circle contains a greater area than any other plane figure bounded by an equal perimeter or outline.

TO FIND CIRCUMFERENCE--

Multiply diameter by 3.1416. Or divide diameter by 0.3183.

TO FIND DIAMETER--

Multiply circumference by 0.3183. Or divide circumference by 3.1416.

TO FIND RADIUS--

Multiply circumference by 0.15915. Or divide circumference by 6.28318.

TO FIND SIDE OF AN INSCRIBED SQUARE--

Multiply diameter by 0.7071. Or multiply circumference by 0.2251. Or divide circumference by 4.4428.

TO FIND SIDE OF AN EQUAL SQUARE--

Multiply diameter by 0.8862. Or divide diameter by 1.1284. Or multiply circumference by 0.2821. Or divide circumference by 3.545.

SQUARE--

A side multiplied by 1.4142 equals diameter of its circumscribing circle. A side multiplied by 4.443 equals circumference of its circumscribing circle. A side multiplied by 1.128 equals diameter } A side multiplied by 3.545 equals circumference } of an equal circle. A side multiplied by 1.273 equals circle inches }

TO FIND THE AREA OF A CIRCLE--

Multiply circumference by one-quarter of the diameter. Or multiply the square of diameter by 0.7854. Or multiply the square of circumference by .07958. Or multiply the square of ¹⁄₂ diameter by 3.1416.

Contents of cylinder = area of end × length. Contents of wedge = area of base × ¹⁄₂ altitude. Surface of cylinder = area of both ends × length × circumference. Surface of sphere = diameter squared × 3.1416, or = diameter × circumference. Contents of sphere = diameter cubed × .5236. Contents of pyramid or cone, right or oblique, regular or irregular = area of base × ¹⁄₃ altitude. Area of triangle = base × ¹⁄₂ altitude. Area of parallelogram = base × altitude. Area of trapezoid = altitude × ¹⁄₂ the sum of parallel sides.

ROMAN TABLE.

I. denotes One. II. „ Two. III. „ Three. IV. „ Four. V. „ Five. VI. „ Six. VII. „ Seven. VIII. „ Eight. IX. „ Nine. X. „ Ten. XI. „ Eleven. XII. „ Twelve. XIII. „ Thirteen. XIV. „ Fourteen. XV. „ Fifteen. XVI. „ Sixteen. XVII. „ Seventeen. XVIII. „ Eighteen. XIX. „ Nineteen. XX. „ Twenty. XXX. „ Thirty. XL. „ Forty. L. „ Fifty. LX. „ Sixty. LXX. „ Seventy. LXXX. „ Eighty. XC. „ Ninety. C. „ One hundred. D. „ Five hundred. M. „ One thousand. X̅. „ Ten thousand. M̅. „ One million.

SOLID MEASURE, OR CUBIC MEASURE.

This is used in measuring bodies, or things having length, breadth and height or depth.

TABLE.

1728 cubic inches (cu. in.) make 1 cubic foot (cu. ft.). 27 cubic feet, „ 1 cubic yard (cu. yd.). 128 cubic feet, „ 1 cord (C.).

CIRCULAR MEASURE.

60 seconds (″) make 1 minute (′). 60 minutes „ 1 degree (°). 360 degrees „ 1 circum. (C.).

The circumference of every circle whatever, is supposed to be divided into 360 equal parts, called _degrees_.

A degree is ¹⁄₃₆₀ of the circumference of any circle, small or large.

A quadrant is a fourth of a circumference, or an arc of 90 degrees.

A degree is divided into 60 parts called minutes, expressed by the sign (′), and each minute is divided into 60 seconds, expressed by (″); so that the circumference of any circle contains 21,600 minutes, or 1,296,000 seconds.

LONG MEASURE--MEASURES OF LENGTH.

12 inches = 1 foot. 3 feet = 1 yard. 5¹⁄₂ yards = 1 rod. 40 rods = 1 furlong. 8 furlongs = 1 common mile. 3 miles = 1 league.

The mile (5,280 feet) of the above table is the legal mile of the United States and England, and is called the statute mile.

Tables of Diameters, Circumferences and Areas of Circles.