ARITHMETIC, GEOMETRY, DRAWING, ETC

Page 7

**48. To divide the circumference into equal parts.** After having drawn the circle, A, (fig. 38), draw two diameters, *d a, b c*, at right angles to each other, dividing the circle into four equal parts. Join the points, *c a*, and divide the line, *c a*, accurately into nine equal parts.

Draw a series of circles concentric with the first, at distances apart equal to one of the divisions of *c a*, and to the number of one, two, three, etc., according as it is required to subdivide the circle, say for a pinion, into seven, eight, nine, etc., equal parts.

With a fine-pointed compass, measure off the radius of the initial circle A. Placing one point at *c*, the other point will give the position of the next leaf, and so on, all around the circumference. If the innermost circle A be selected for subdivisions, six divisions will be obtained, and there will be one more division for each larger circle

The operation will be facilitated by selecting the first circle, so that the line *a c* contains exactly nine divisions equal to those of some scale that is easily accessible. Such a circle can be easily found, by first drawing the two diameters, laying the scale in the direction *c a*, and determining by trial the radius for which the first and ninth divisions correspond to *a* and *c* respectively.

**49. To divide a surface into rings of equal or proportional superficial area.** The following solution is due to M. Brocot:

Let *a d* be the radius of a circle (fig. 39), that is required to be subdivided into four rings of equal area by concentric circles. Taking *a d* as a diameter, draw the semi-circumference, *a b d*; accurately divide *a d* into four equal parts, and at each point so obtained, erect a perpendicular. Through the intersections of these perpendiculars with the semicircle, draw a series of concentric circles; they will trace out rings, 1, 2, 3, 4, that have equal superficial areas.

If it be required to divide the surface in a given proportion, divide the line *a d*, according to that proportion.

The right-hand side of (fig. 39) gives a special application of this method to the division into two equal areas of the interior of a barrel exclusive of the space occupied by the arbor-nut. If the mainspring accurately covers *i c* when wound up, and *i j* when unwound it will give the greatest possible number of turns.

**50.** Solar time is taken from the revolutions of the earth, and the watchmaker can easily get the exact solar time of any point at which he may happen to be by a little calculation from known standards. These standards are; 1. The zenith. 2. The longitude of the point of observation. 3. The difference between noon at the point of observation and noon of a known meridian either east or west of the point of observation. The zenith is that point in the heavens where the rays of the sun are in a plane exactly perpendicular to the surface of the earth at the point of observation, and when the rays of sunlight are in this plane it is noon at that point. The circumference of the earth is divided into 360 degrees or meridians of longitude, so that as the earth revolves once every twenty-four hours, each of these meridians will pass the zenith, or fixed point, in that time. In twenty-four hours there are 24×60=1,440 minutes, so that the interval between the passage of one meridian and the next will be 1,440÷360=4 minutes. A degree of longitude is divided like an hour, into minutes and seconds, so that;

1 degree of longitude equals 4 minutes of time.

1 second of longitude equals 4 seconds of time.

1 second of longitude equals 1/15 or .066 seconds of time.

**51.** Thus it happens that, at a town one degree east of a given point the sun will be visible four minutes sooner, and if to the west, four minutes later than at that point. The "local," or solar time, therefore, will be four minutes earlier at the first town, and four minutes later at the second town, than it is at the point of observation.

**52.** It will be readily seen that, having any two of the three factors given above, the other can be readily found. Thus having the time of a given meridian and the local noon or meridian time, the longitude can be readily found; or, having the longitude (which can be readily obtained from a surveyor) and the time of a given meridian, "noon," can be calculated, etc. The first method is followed in calculating distances at sea; the chronometer keeping Greenwich time, and the local noon giving the longitude.

When great accuracy is necessary, however, a fixed star is used as a means of observing the exact time when a revolution of the earth is completed, as the revolution of the sun in its orbit causes a slight variation during the year. For further information on this point the reader is referred to works on astronomy.

To obviate the constantly varying time in running east or west, the railroads use the time of a given meridian over each fifteen degrees of longitude, and as each degree of longitude equals four minutes of time, it follows that only the hour is changed in changing from one standard to another. In Europe the zero longitude, or the time of the meridian of Greenwich is used. In the United States the time of the 75th degree, which passes through Philadelphia, is used from the 67th to the 80th degree, which comprises the territory from Princeton, Maine, to a line drawn north and south, passing through Erie and Pittsburgh, Pa., and is called Eastern Time. The time of the 90th meridian is used from the 80th to the 102nd meridian, and is called Central time. The time of the 105th meridian is used from the 102nd to the 114th meridian, and is called Mountain time. The time of the 120th meridian is used from the 114th meridian to the coast (which ends at about the 124th meridian) and is called Pacific time. The time of the various standards is telegraphed through their various territories at noon each day, and furnishes an accurate standard of comparison to all watch-makers.

**53.** In very many cities the actual or solar noon has been discarded and the railway standard adopted, thus making but one standard and removing the source of confusion and annoyance to many people. In others, however, the two standards are still used, and it becomes necessary for the watch-maker to be able to calculate both standards, in case of accident or irregularity in his regulator. Hence he should calculate his longitude within one second by means of the difference between railroad and local noon, and have the nearest surveyor correct his reckoning; then, by means of the accurate longitude and the railroad time, correct the solar time; then by means of the solar noon and the longitude calculate the railroad time. When all these calculations check each other perfectly, he possesses all the time data he needs for that place, and can correct his standard or regulator if at any time it should become irregular. The calculations are very simple, and can be easily performed from the data given above.

Submitted by: Samuel Kirk (##)