ARITHMETIC, GEOMETRY, DRAWING, ETC

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**23.** It is needless here to describe the rule, set-square, T-square, bow-compass, etc., as every one knows them.

*To verify the accuracy of a rule.* On a perfectly flat smooth surface carefully draw, with the rule in question, a fine straight line. Then turn the rule over, hinging it as it were on the line just drawn; if quite straight the edge of the rule will exactly coincide with the line, in this new position, throughout its entire length. Each edge should be thus examined.

*To verify the accuracy of a set-square.* Having fixed an accurate rule on a smooth surface, place one edge of the set-square against it, and draw a line along the edge perpendicular to the rule; then, having turned the set-square, hinging it on the line just drawn, bring it against the rule and along the line. If the square is true the edge and line will coincide throughout their length.

**24. The Protractor.** Fig. 7 represents a common form of this instrument. It is made of horn, or, it of metal, the inner portion is cut away, leaving only a base and a semicircular arc, which is divided into 180 equal parts or degrees; a complete circle would therefore consist of 360 such degrees. The point indicating the center of the arc, should be very small in order to facilitate the exact setting of it at the apex of an angle.

When an angle has to be drawn with accuracy, the protractor is unsuitable; it will be better to adopt one of the methods described at paragraph **37,** or trigonometrical methods.

**25. Drawing scales.** When an object is represented by a drawing, if the dimensions are the same as those of the object itself, or, rather, as they would project on to a horizontal or vertical plane, the drawing is said to be full size; but the object is generally represented either on an increased or diminished scale, which is defined, the proportions between all the parts being still, however, maintained the same.

With a view to avoid the many calculations that such a change would involve, it is usual to employ drawing scales. The following notes will sufficiently explain their construction and use.

Let it be required to reproduce a large drawing on a small scale, in such a proportion that the dimensions are reduced in the ratio of 10 to 1.

Take a straight line of indefinite length, *a b,* fig. 8, and mark out on it spaces equal to *a* 1, which represents any measurement taken on the original object; at *c*, the 10th division, draw a perpendicular, and on it measure *c g* equal to *a* 1, or one-tenth of *a c* and join *a g*.

Through the points indicating the divisions into tenths draw lines parallel to *c g*, and you will thus have a series of triangles, *d a*1, *d′ a* 2, d″ a 3, etc., similar to the triangle *g a c*. In virtue of a well-known property of such triangles (**15**), *d* 1 will be one-tenth of *a* 1; *d′* 2 one-tenth of *a* 2; and so on.

Thus, if a measurement taken on the object, or on a large drawing, is equal to *a x*, it will only be needful to turn the compass on the point *x* as a center, and to observe accurately the perpendicular height, *x z*, to ascertain the corresponding measurement on the reduced scale.

Such a scale can be employed to measure meters and deci-meters, or feet and inches (but in this latter case, it would have been necessary to mark off 12 instead of 10 divisions from *a*). Since *a* 1 might be made to represent one metre; *a* 2, two meters, etc., in virtue of the principle of the triangle already referred to, *d* 1 will be the tenth of *a* 1, and will therefore represent a decimeter; *d′* 2 will represent 2 deci-meters, etc. The length required to represent, say 5.3 will be ascertained by taking the distance *a* 5, to which the distance *d″* 3 is added. Similarly 6 feet 2 inches would be given by *a* 6, to which *d′* 2 is added on a 12-division scale.

**26** The following description of one of these decimal scales, which is engraved on metal or ivory, and often included in cases of drawing instruments, will suffice to enable any one to construct a scale on this principle, that goes to a still further degree of accuracy, measuring for example, meters, deci-meters and millimeters, or yards, feet and inches.

Let A B, fig. 9, be a flat rectangular rule, divided throughout its length by parallel equidistant lines into ten strips. At right angles to these are the lines *o o′, a a′,* etc., separated from one another by a distance of one centimeter (doubled in the drawing in order to make the details more clear.) The first centimeter is subdivided along the two edges, A *o, n′ o,* into 10 equal parts or millimeters, and the division, *o,* on the upper edge is joined by an oblique line with the 1 on the lower edge, and the others by parallel oblique lines as shown in the figure. Thus *c i* will be one-tenth of a millimeter, *s j* two-tenths, and so on.

If the compass is opened so as to reach from *x* to *z*, it will be seen that it covers a space of 16 millimeters and 2-10ths of a millimeter, for there are one large division (or 10 mm.), 6 smaller divisions (or millimeters) plus a fraction of a millimeter equal to *s j* or 2-10ths of a millimeter.

**27. Sector.** When it is required to reduce the scale of a drawing, subject to the condition that the dimensions shall be all diminished in the ratio of two given lines, we may state the problem thus:

The longer of the two given lines is to the shorter, as any given dimensions of the old drawing is to *x*. The value of *x* thus determined will be the corresponding dimension of the new figure.

Such a rule of three proposition would involve a considerable amount of work, and the required result can be arrived at with greater facility by the geometrical methods which forms the basis of the scale just described, or, better still, by using the sector shown in fig. 10. It consists of two brass or ivory legs hinged about a center *m* which is at the apex of the angle *n m c* formed by two straight lines similarly divided into equal parts.

It is employed as follows: Let us assume that a drawing has to be reduced in the ratio of the line A to the line B; set off the length A along *m n,* and suppose its extremity to be at *s,* where division number 5 occurs. Open a compass to a distance equal to B, and placing one point on *s,* open the two legs of the scale until the second point coincides exactly with the corresponding division *t,* that is, with the 5 on the other leg, *m c.* Maintaining the scale open to this amount, it is only needful, after measuring a distance on the original drawing or object, to set it off along *m n,* and to measure the distance between its extremity and the corresponding point on the other leg; this distance will be the dimension on the reduced scale.

**28. Proportional compass.** This consists of two equal stems terminating with points, fig 11. They are cut through for a portion of their length, and provided with a slide forming a hinge, that can be clamped by a screw *a* in any position. Graduations on the two slots and a mark on the slide indicate in what position of the slide *a*, the length *ab* (equal to *ag*) is equal to 1/2, 1/3, 1/4, etc., of *ad;* and thus show what is the ratio of *gb* to *cd,* a ratio which is independent of the extent to which the arms are opened.

**29. The vernier.** The vernier consists of a small graduated slide which is adapted to a graduated rule or circular arc with a view to ascertain the value of small fractional parts of the divisions marked on the rule or arc.

Let A B, in fig. 12, be a rule divided into millimeters (the proportions are enlarged in the drawing so as to avoid confusion among the lines), and let it be required to determine a length to within the tenth of a millimeter.

As the measurement is required to the tenth, take ten less one or nine of the divisions of the scale; they will extend from 0 to IX, and this represents the acting length of the vernier.

Subdivide the vernier into ten equal parts; it is manifest that each graduation of the vernier differs from the original subdivisions of the rule by 1-10th of a graduation of the latter. In other words, unity on the vernier is equal to 9-10ths of unity on the rule.

When the rule and vernier are placed as shown in fig. 12, so that the 0 on both scales coincide, the successive divisions on the rule (marked with Roman numerals for distinction) will be progressively more and more in advance of the corresponding divisions on the vernier in the following proportions:-

The marks I and 1 are 1-10th apart; the marks II and 2, 2-10ths; III and 3, 3-10ths; and so on, The mark X being 10-10ths, or one complete division in advance of 10, this division being a unit on the scale.

Thus if the vernier is caused to slide along the edge of the rule, when 1 and I are 1-10th; when 2 coincides with II, it has advanced 2-10ths; and so on.

Let it be required to determine the distance P *d*, fig. 13. The division 6 on the vernier coincides with a division of the scale; hence it follows that the extremity *d* of the vernier is at a distance of 6-10ths millimeters from III, the next division of the scale to the left. The distance between P and *d* is thus 3.6 millimeters.

With a vernier showing tenths, if two consecutive divisions of the vernier fall between two divisions on the rule, and there does not appear to be a tendency towards one side more than towards another, even when observed with a strong glass, it is possible to take an approximate reading to the twentieth.

In measuring circular arcs a curved vernier is used in place of a straight one, and its graduations are made correspond with those on the circle as shown in fig. 14.

**30. Micrometer screw.** By employing a micrometer screw it is possible to measure infinitesimal amounts, but the screw must be perfectly accurate, and must work without appreciable backlash or loss of time.

Assume V, fig. 15, to be such a screw, having a pitch of 1 millimeter. It will advance by this amount with each complete rotation.

To the head of the screw is attached a disc of such a size that its rim can be divided into a number of equal parts, say a hundred. These graduations may be marks on the edge or notches cut in it when an index is required to stop in them; but the index is less frequently met with than a simple divided straight-edge almost in contact with the disc. The divisions round the disc are numbered in ascending order as the points *c* and *a* separate, so that zero comes under the index or rule when these points are in contact. Readings of the numbers will thus afford a measure of the displacement of the point of the screw.

When the disc is rotated the point *a* will move towards or from *c* by 1-100th of a millimeter for each division passing under the straight-edge, and one millimeter for each complete rotation. It is thus possible to obtain the dimensions of an object when it enters without play between the two jaws to within an error of about 1-100th lf a millimeter if the instrument is accurately made.

If, instead of passing the object between the two jaws, it is gripped by them, the measurement will be less exact, as no account is taken of the pressure exerted and of the elasticity. (**44.**)

Submitted by: Samuel Kirk (##)