Chapter V: Part 5

Each applicant for a patent is required by law to furnish a drawing of his invention whenever the nature of the case admits of it. The drawing must be signed by the inventor or the name of the inventor may be signed on the drawing by his attorney-in-fact, and in either case must be attested by two witnesses. The drawing must show every feature of the invention covered by the claims.

When the invention consists of an improvement on an old machine, the drawing must exhibit, in one or more views, the invention proper, disconnected from the old structure, and also, in another view, so much only of the old structure as will clearly show the connection of the invention with the old machine.

Several editions of the patent-drawings are printed, the smallest of which is about 3 × 4³⁄₄ inches, so that the drawing must be so made that it will stand a reduction of about one-fourth. This work is done by the photo-lithographic process, and therefore the character of the original drawing must be brought as nearly as possible to a uniform standard of excellence suited to the requirements of the process.

NOTE.--These rules will be found most useful to many readers of this
work--hence their introduction at this point. Nearly 50,000 patents
are “applied for” in the United States every year.

The following rules are given by the Patent Office for guidance:

1. Drawings must be made upon pure white paper of a thickness corresponding to three-sheet Bristol board. The surface of the paper must be calendered and smooth. India ink alone must be used, so as to secure perfectly black and solid lines.

2. The size of a sheet on which a drawing is made must be exactly 10 × 15 inches. One inch from its edges a single marginal line is to be drawn, leaving the “sight” precisely 8 × 13 inches. Within this margin all work and signatures must be included. One of the shorter sides of the sheet is regarded as its top, and measuring downwardly from the marginal line, a space of not less than 1¹⁄₄ inches is to be left blank for the heading of title, name, number and date.

3. All drawings must be made with the pen only. Every line and letter, signature included, must be absolutely black. This direction applies to all lines, however fine, to shading, and to lines representing cut surfaces in sectional views. All lines must be clean, sharp, and solid, and they must not be too fine or crowded. Surface shading, when used, should be open. Sectional shading should be made by oblique parallel lines about ¹⁄₂₀ of an inch apart. Solid black should not be used for sectional or surface shading.

4. Drawing must be made of the fewest lines possible, consistent with cleanness. The plane upon which a sectional view is taken should be indicated by a broken or dotted line. Heavy lines on the shade side of objects should be used, except where they tend to thicken the work and obscure letters of reference. The light is always supposed to come from the upper left hand corner at an angle of 45 degrees.

5. The scale to which a drawing is made should be large enough to show the mechanism without crowding. The number of sheets used must never be more than is absolutely necessary.

6. The different views should be consecutively numbered. Letters and figures of reference must be carefully formed. They should, if possible, measure at least one-eighth of an inch in height.

If the same part of an invention appears in more than one view of the drawing it must always be represented by the same character.

7. The signature of the inventor is to be placed in the lower right-hand corner of each sheet, and those of the witnesses at the lower left-hand corner.

The title should be written with pencil on the back of the sheet.

Drawings should be rolled for transmission, never folded.

On page 235, fig. 292 exhibits a reproduction of a patent office drawing, used in connection with specification papers in an application for a United States patent.

ENGLISH PRACTICE.

The rules for patent drawings in England are practically the same as in the United States; the paper sizes are, however, different. They must be on sheets of one of the two following sizes (the smaller being preferable), 13 inches at the sides by 8 inches at the top and bottom, or 13 inches at the sides by 16 inches at the top and bottom, including margin, which must be one-half an inch wide.

If there are more figures than can be shown on one of the smaller-sized sheets, two or more of these sheets should be used in preference to employing the large size. When an exceptionally large drawing is required, it should be “_continued_” on subsequent sheets. There is no limit to the number of sheets that may be sent in.

Useful Hints and “Points.”

Many of these “points” are repetitions, with but little variation from
the way they have been previously stated; they are thus repeated to
emphasize their practical worth.

A good draughtsman leaves his work in such a state that any competent person can without difficulty ink in what he has drawn.

The criterion of a good set of drawings is that with a properly prepared specification they are complete in themselves and require no explanation.

A “break” in a figure or object in a drawing is shown in rough irregular lines, as in fig. 134, on page 131; this is useful when the paper is not large enough to show the whole.

Never use a sloping line in writing fractions on a drawing. The objection arises from the fact that such a dimension as 1³⁄₁₆, if written with the inclined line, unless very distinctly executed, may be read as ¹³⁄₁₆.

In inking do not draw the lines further than you wish them to go, but in penciling it is well to extend the lines, free up.

Never use a scale for a ruler.

Do not overload the pen with ink.

Having filled the pen, nearly close the nibs and try the width of the line on a piece of paper or the margin of the drawing.

Never refill or lay the pen aside without first cleaning it.

The application of the science of geometry to the drawing-board is absolutely necessary to success, for the reason that the whole fabric of mechanical drawing rests on the principles of geometry, which is well termed the science of measurements.

Section lines should be the last inked and always without previous penciling.

Center lines are necessary in working drawings.

In choosing T-squares, care should be exercised to see that the head slides up and down the _left_-hand side of the board easily, and that when pressed against the board with the left hand there is no “slogging” of the blade up or down, or in other words, that the head is bearing firmly for its whole length against the board.

The best place for the title of a drawing is said to be the upper left-hand corner; this facilitates the filing of the sheet.

Never use a soft pencil except for finishing in shadow lines.

The rubber should always be kept clean.

Great care should be taken to keep drawing boards out of the way of heat or damp, as these cause the wood to warp.

Circles and curves are to be “inked in” before straight lines. First ink the smallest and afterwards the larger curves.

Do not press heavily on the pencil so as to cut the paper, but draw lightly, so that the mark can be erased and leave no trace, especially if the drawing is to be inked.

The draughtsman should commence his work at the top of the paper, keeping the lower part covered over until he needs to use it.

Shade lines should be avoided in all working drawings, as their use interferes with accurate measurements.

To make ink stick to the tracing cloth, with a woolen cloth rub some powdered chalk or pounce over the surface on which the ink lines are to be drawn, then wipe the surface clean and use a good quality of ink.

For striking small circles a small bow pen should be used.

To fix lead pencil marks on sketches so that they cannot be readily erased, sponge them with milk carefully skimmed, then lay blotting paper over them and iron with a hot flat-iron.

To have the ink preserve its fluidity and to keep out all dirt and dust, keep the cover on the ink slab; the mistake is often made of putting too liberal a supply of water in ink well, which causes a waste of both time and ink; no more should be prepared than to meet immediate requirements.

Always draw on the right side of the sheet, which can be found by holding the sheet up to the light and looking across its surface with the eye nearly in the same plane as the paper; note which side is the smoothest and has the least number of blemishes on it; this is the right side to draw on.

As to sharpening pencils, it is always best to cut a chisel point on the pencil used for drawing, and put a circular point on the pencils in the bow pencil and pencil leg. The chisel point makes a finer line and lasts much longer than a round point.

The varnish used in many large drawing-rooms is simply white shellac dissolved in alcohol; it requires a little experience to mix these to a proper consistency, but this is soon acquired.

Never sharpen your pencil over the drawing.

A center line of a drawing is the line upon which the figure is to be constructed; the center line is the first line to be drawn.

The T-square belongs to the left side of the drawing-board, and is operated by the left hand. The right hand should be kept free for the purpose of picking up pencil, pen and bows, adjusting and marking off. The left hand controls the T-square and the triangle that slides along the upper edge of the square; the right hand is for the instruments.

The advantage of a paper rule or scale is that the paper will expand and contract under varying degrees of atmospheric moisture the same as the drawing does.

Avoid rubbing out and constantly cleaning the drawing with India rubber; if wrong lines are made or it is desired to make alterations, the part to be changed should be rubbed out and completely re-drawn.

When using the bows see to it that the steel-pointed leg that is put down first on the paper, to secure a center for a curve or a circle, is a trifle longer than the pencil or pen leg.

To clearly indicate the position of a center which is to be used again, lightly pencil a small circle about it; never put the point of a pencil in the center hole to enlarge or blacken it; the prick point made by the dividers and needle points should be no more than can be just seen, hence the circle to be made as advised above.

Be particular in having the legs of the dividers exactly the same length, and sharp, so that in pricking off distances, and dimensions, and centers, the indent or hole made in the paper is as small as possible.

The term “plane” means a perfectly flat surface; that is, something which has length and breadth but no thickness.

The best way to indicate on the drawing the surfaces which are to be finished is to write on the lines which represent the finished surfaces “finished,” tool-finish, or “faced,” according to the degree of finish required. The single letter _f_ is frequently used.

Avoid fingering the drawing sheet as much as possible; in pointing to any part of the drawing use a pencil and not the finger.

Remember that a drawing is made to be read.

The skill in inking does not depend on the fineness of the line, but on its clearness.

A soft pencil should never be used on a mechanical drawing unless in rare cases when it is used for pencil shading; the hardness or softness of pencils is denoted by letters.

Never ink any portion of a drawing until the penciling is entirely finished.

Stretching or pasting the paper to the board is very seldom resorted to, for the reason that the mechanical drawings are _to scale_ and the paper is natural when pinned to the board and more correct than if under a strain. Mechanical drawings are always required in practice _right away_, and time would be wasted and lost in damping and pasting and drying again.

A working drawing, whether made to a scale or not, must have all the dimensions plainly written upon it, for a workman should never be compelled to measure a drawing.

In marking off distances, centers, etc., a fine needle point is useful; the hole should not be punctured through the paper, merely a prick point, so that it will leave an impression, which will not be obliterated by the use of rubber; drawing-pens are often equipped with such a needle point in the end of the handle, that is visible only when the pen is unscrewed from the handle; but in the absence of one of this kind the point of the divider leg will be of use.

Mechanical construction drawings represent a large amount of mental and manual work, as well as a considerable cost in money; hence, they are of value quite as much as property which has been acquired by the expenditure of either labor or capital. It is wise to keep copies of original designs and sketches, as well as data and formulæ, for record and comparison.

The best system for keeping drawings is to make them of certain standard sizes, and to keep them flat, unrolled, in drawers, numbered, lettered and labeled.

In an office where space is limited and drawings have to be rolled it is well to use a number of pasteboard cases about three feet long and three inches in diameter. These are shown in fig. 294.

A puncture can be made near the top and, when a new drawing or blue-print is inserted in this cylindrical case, a cardboard tag can be looped through the puncture. This label will give the title and number of drawings in that case.

A manuscript book methodically and neatly kept should tell immediately the number of the drawing and the case.

Fig. 293 is good for practice in line drawing and also as an optical illusion. “You look and are deceived. At first glance you say, ‘Of course, those two lines are curved.’ You are mistaken. They are exactly parallel. In order to prove this hold them up edgewise to the eye. It is, of course, the subsidiary lines which lead the vision astray. It is a case of first impressions being quite wrong.”

Linear Perspective.

It should be mentioned that this subject is outside the limits of mechanical drawing, which only deals with objects that can be measured, projected or dimensioned to an accurate scale.

But, in rounding out the more formal subjects it is well to look a little outside the rigid lines of mechanics into the methods of nature, for no system of teaching drawing is complete that does not include some explanations for sketching from nature--the objects being always around the student, the eye always clear to see and the hand only needing the training to make permanent the impressions received.

The word perspective means to _see through_; the word perspective being derived from the Latin word _perspicere_, to look through, hence, perspective is a science which teaches us to _see_ correctly and enables us to represent the _appearance_ of anything we may wish to draw; care should be taken in perspective drawing, to select objects interesting in themselves, and the best specimens of their class, so as to cultivate taste, while they at the same time afford useful and instructive drawing lessons.

The meaning of the term linear perspective is a line view; the previous examples have been composed of surfaces placed fronting the eye; perspective is the science which treats of the changes of form produced by viewing them in various oblique positions.

The slightest alteration of _position_ will change the _appearance_ of an object; this can be easily shown--for illustration take a coin, the actual shape of which is a perfect round; or, strictly speaking, a circle. If we take the coin between the thumb and the first finger, holding it in an upright position, and exactly facing the eyes, as in Fig. 296, it appears of its true form, viz., a circle. If we alter its position, balancing it upon the thumb, in a level position, with its edge directly opposite the eye, as in Fig. 297, its appearance is changed, and what we know to be really a circle, appears to us as a straight line.

Now, still balancing the coin upon the thumb, but changing its position with regard to the eye, by holding it a little lower than in the last position, that is slightly beneath the level of the eye, as in fig. 298, we see both the edge and the surface, the coin now appearing neither a circle nor a straight line, but a curved figure of an elliptical form. Thus the same coin held in three different positions has assumed three different shapes.

Let us take two coins of the same size, holding (in the position shown at fig. 296) one in each hand. Now, closing one eye, (which will make the experiment more clear), hold one coin out at arm’s length, and the other at about the distance of a foot from the eye. On comparing them, we find that the coin which is further from the eye appears less than the nearer one. We know that the coins are really equal in size, yet one appears smaller than the other.

We thus see that when we change the position of an object, we have as a consequence a change of appearance; also that the change of appearance may affect both the shape and the size of the object.

These diversities of appearance may be remarked in everything around us. We can observe them in the street by looking at a building from different points of view, or by comparing the apparent sizes of the street lamps; in the railway station, by watching the arriving or departing train; and at sea, by noticing the vessels as they approach, or as they retire, ultimately vanishing from our sight in that line where the sea and sky appear to meet.

All these interesting variations of appearance are in strict accordance with the laws of =GEOMETRY= and =OPTICS=. The former subject has been enlarged upon beginning with page 81 of this work, where a line, a point, an angle, etc., are defined; other terms are explained at page 41 and the following pages; to these we add a few definitions essential to the subject.

=A PLANE= is a surface which is perfectly even and flat; to use a familiar illustration, a plane is like the surface of a sheet of plate glass; recollect particularly, that a surface which is at all curved, is not a plane.

The =GROUND-PLANE= is the plane on which we stand; the _base-line_ is an imaginary line passing through the middle of the feet as we stand square and erect; and the _vertical plane_ is supposed to stand on the base-line and perpendicular to it.

Planes are parallel to each other when they are throughout their entire surfaces the same distance apart.

=THE PERSPECTIVE PLANE= is an upright square of glass, usually framed like a picture, with a base, so that it can stand up alone. This is placed between the eye of the spectator and the subject to be drawn, and as the drawing is sometimes made directly upon it, it is sometimes called the _Picture_ or the _Plane of the Picture_.

=HORIZONTAL= means perfectly level, like the surface of still water. We must be careful to understand perfectly the difference between the terms “level” and “even” or “flat.” A surface may be even or flat, without being level. Thus the wall is even and flat, but it is upright, not level; level means a fixed, constant position.

In fig. 301 a house is shown in perspective in which the line _H L_ is the line of the horizon and _V P_ is the,--

=VANISHING POINT.=--The vanishing point is familiarly represented by the rails on a trolley track on a straight road, which seem to approach each other in the distance, as shown in fig. 302 at _V P_.

_All parallel lines seen in perspective appear to meet in the same vanishing point._

The value of the vanishing point may be seen in the view of a wooden house, fig. 303, where it (_V P_) gives direction to the retiring lines of the roof, side planks and door.

=POINT OF SIGHT.=--This is that point in the eye where the lines or rays from the object cross each other, as shown at _P_ in fig. 305, also in fig. 299 at _S_.

=VERTICAL= means perfectly upright. If we attach a piece of thread to a weight, a small piece of lead for example, and hold the thread with the lead hanging downwards, the thread will fall in an upright or vertical position.

=PARALLEL= lines are said to be parallel to each other when they are throughout their whole lengths the same distance apart.

=PERPENDICULAR.= When one straight line, meeting another, makes the angles at the point of contact equal, each of the angles is called a right angle, and the lines are said to be perpendicular to each other. Remember especially that perpendicular and vertical have not the same meaning. Vertical means an unvarying upright position. Perpendicular means that one line or plane meets another line or plane at right angles.

The fig. 295 on page 246 is a study in perspective, showing a water reflection. As rays from every visible part of the object are reflected, all following the same law, the reflection will appear to the eye _inverted_, and of the _same size as the object_. The arch itself forms the upper half of a hollow cylinder, and the reflection forms the lower half. The reflection shows much more of the interior of the arch than can be seen directly. The leaning tree, the boy fishing, and the receding banks, all are seen in accordance with the laws of reflection and perspective.

=THE HORIZONTAL LINE, THE POINT OF SIGHT AND THE VANISHING POINTS= are the principal items. These should be studied in every room and during every walk, and the more pleasing accidents of form stored in the mind or committed to paper for future use.

=OPTICS=, the science of sight, gives us the following laws:

1. That we see by the agency of light.

2. That light passes from objects to our eyes.

3. That light travels in straight lines, which are called Visual Rays.

The human eye may be briefly described as a chamber of a spherical or globular form, with a circular opening in front. This circular opening is called the pupil, and through it the visual rays pass to the interior of the eye. The visual rays, passing from space in all directions through the small pupil, are received upon what may be called the interior wall of the globular chamber forming the eye (see fig. 305). This interior wall is called the retina, and upon it the impressions of external objects are received, just as they are received upon a screen in a dark chamber. These impressions are conveyed by the optic nerve from the retina to the brain.

In front of the pupil is a segment of a small sphere, composed of the cornea and the aqueous humor, both of which are transparent, and from their shape and density have a convergent effect upon the rays passing through them.

Behind the pupil is the transparent crystalline lens, which, from its shape and its elasticity, is a powerful agent in aiding the convergence of the rays, and in bringing objects at various distances to a clear focus upon the retina.

The pupil has the power of contraction and dilation, which is influenced by the quantity of light entering the eye, but when it is dilated to the utmost its size is very small in comparison with the great chamber forming the body of the eye.

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.

AND

INDEX]

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.