CHAPTER I.

ON MICROMETERS.

A micrometer is an instrument attached to a telescope, in order to measure small spaces in the heavens, such as the spaces between two stars, and the diameters of the sun, moon and planets—and by the help of which the apparent magnitude of all objects viewed through telescopes may be measured with great exactness.

There are various descriptions of these instruments, constructed with different substances, and in various forms, of which the following constitute the principal variety. The Wire micrometer—the Spider’s line micrometer—the Polymetric reticle—Divided object glass micrometer—Divided eye-glass micrometer—Ramsden’s Catoptric micrometer—Rochon’s crystal micrometer—Maskelyne’s Prismatic micrometer—Brewster’s micrometrical telescope—Sir W. Herschel’s Lamp micrometer—Cavallo’s Mother of Pearl micrometer, and several others. But, instead of attempting even a general description of these instruments, I shall confine myself merely to a very brief description of Cavallo’s Micrometer, as its construction will be easily understood by the general reader, as it is one of the most simple of these instruments, and is so cheap as to be procured for a few shillings; while some of the instruments now mentioned are so expensive, as to cost nearly as much as a tolerably good telescope.[37]

This micrometer consists of a thin and narrow slip of mother of pearl finely divided, which is placed in the focus of the eye-glass of a telescope, just where the image of the object is formed; and it may be applied either to a reflecting or a refracting telescope, provided the eye-glass be a convex lens. It is about the 20th part of an inch broad, and of the thickness of common writing paper, divided into equal parts by parallel lines, every fifth and tenth of which is a little longer than the rest. The simplest way of fixing it is to stick it upon the diaphragm which generally stands within the tube, and in the focus of the eye-glass. When thus fixed, if you look through the eye-glass, the divisions of the micrometrical scale will appear very distinct, unless the diaphragm is not exactly in the focus of the eye-glass, in which case it must be moved to the proper place;—or, the micrometer may be placed exactly in the focus of the eye-lens by the interposition of a circular piece of paper, card, or by means of wax. If a person should not like to see always the micrometer in the field of the telescope, then the micrometrical scale, instead of being fixed to the diaphragm, may be fitted to a circular perforated plate of brass, of wood, or even of paper, which may be occasionally placed upon the said diaphragm. One of these micrometers, in my possession, which contains 600 divisions in an inch, is fitted up in a separate eye-tube, with a glass peculiar to itself, which slides into the eye-piece of the telescope, when its own proper glass is taken out.

To ascertain the value of the divisions of this micrometer.—Direct the telescope to the sun, and observe how many divisions of the micrometer measure its diameter exactly. Then take out of the Nautical Almanack the diameter of the sun for the day on which the observation is made. Divide it by the above-mentioned number of divisions, and the quotient is the value of one division of the micrometer. Thus, suppose that 26½ divisions of the micrometer measure the diameter of the sun, and that the Nautical Almanack gives for the measure of the same diameter 31´: 22´´, or 1882´´. Divide 1882 by 26.5, and the quotient is 71´´ or 1´: 11´´, which is the value of one division of the micrometer; the double of which is the value of two divisions, and so on. The value of the divisions may likewise be ascertained by the passage of an equatorial star over a certain number of divisions in a certain time. The stars best situated for this purpose are such as the following—δ in the Whale, R. A. 37°: 3⅓´, Dec. 37´: 50´´ S; δ in Orion, R. A. 80°: 11´: 42´´, Dec. 28´: 40´´ S; υ in the Lion, R. A. 171°: 25´: 21´´, Dec. 23´: 22´´ N.; η in Virgo R. A. 182°: 10´, Dec. 33´: 27´´ N. But the following is the most easy and accurate method of determining the value of the divisions:—

Mark upon a wall or other place the length of six inches, which may be done by making two dots or lines six inches asunder, or by fixing a six inch ruler upon a stand. Then place the telescope before it, so that the ruler or six-inch length may be at right angles with the direction of the telescope, and just 57 feet 3½ inches distant from the object-glass of the telescope; this done, look through the telescope at the ruler, or other extension of six inches, and observe how many divisions of the micrometer are equal to it, and that same number of divisions is equal to half a degree, or 30´; and this is all that is necessary for the required determination. The reason of which is, because an extension of six inches subtends an angle of 30´, at the distance of 57 feet, 3½ inches, as may be easily calculated from the rules of plane Trigonometry.

figure 85.

Fig. 85, exhibits this micrometer scale, but shows it four times larger than the real size of one which was adapted to a 3 feet achromatic telescope magnifying 84 times. The divisions upon it are the 200ths of an inch, which reach from one edge of the scale to about the middle of it, excepting every fifth and tenth division, which are longer. Two divisions of this scale are very nearly equal to one minute; and as a quarter of one of these divisions may be distinguished by estimation, therefore an angle of ¹/₈ of a minute, or of 7½´´ may be measured with it. When a telescope magnifies more, the divisions of the micrometer must be more minute. When the focus of the eye-glass of the telescope is shorter than half an inch, the micrometer may be divided with the 500ths of an inch; by means of which, and the telescope magnifying about 200 times, one may easily and accurately measure an angle smaller than half a second. On the other hand, when the telescope does not magnify above 30 times, the divisions need not be so minute. In one of Dollond’s pocket telescopes, which, when drawn out for use is only 14 inches long, a micrometer with the hundredths of an inch is quite sufficient, and one of its divisions is equal to little less than 3 minutes, so that an angle of a minute may be measured by it. Supposing 11½ of those divisions equal to 30´ or 23 to a degree—any other angle measured by any other number of divisions, is determined by proportion. Thus, suppose the diameter of the sun, seen through the same telescope, be found equal to 12 divisions, say As 11½ divisions : are to 30 minutes :: so are 12 divisions : to ((12 × 30)/11.5) 31.3, which is the required diameter of the sun.

Practical uses of this Micrometer.—This micrometer may be applied to the following purposes:—1. For measuring the apparent diameters of the sun, moon, and planets. 2. For measuring the apparent distances of the satellites from their primaries. 3. For measuring the cusps of the moon in eclipses. 4. For measuring the apparent distances between two contiguous stars—between a star and a planet—between a star and the moon—or between a comet and the contiguous stars, so as to determine its path. 5. For finding the difference of declination of contiguous stars, when they have nearly the same R. Ascension. 6. For measuring the small elevations or depressions of objects above and below the horizon. 7. For measuring the proportional parts of buildings, and other objects in perspective drawing. 8. For ascertaining whether a ship at sea, or any moving object is coming nearer or going farther off; for if the angle subtended by the object appears to increase, it shows that the object is coming nearer, and if the angle appears to decrease, it indicates that the object is receding from us. 9. For ascertaining the real distances of objects of known extension, and hence to measure heights, depths, and horizontal distances. 10. For measuring the real extensions of objects when their distances are known. 11. For measuring the distance and size of an object when neither of them is known.

When the micrometer is adapted to those telescopes which have four glasses in the eye-tube—and when the eye-tube only is used, it may be applied to the following purposes:—1. For measuring the real or lineal dimensions of small objects, instead of the angles. For if the tube be unscrewed from the rest of the telescope, and applied to small objects, it will serve for a microscope, having a considerable magnifying power, as we have already shown, (p. 348); and the micrometer, in that case, will measure the lineal dimensions of the object, as the diameter of a hair, the length of a flea, or the limbs of an insect. In order to find the value of the divisions for this purpose, we need only apply a ruler, divided into tenths of an inch, to the end of the tube, and, looking through the tube, observe how many divisions of the micrometer measure one tenth of an inch on the ruler, which will give the required value. Thus, if 30 divisions are equal to ¹/₁₀th of an inch, 300 of them must be equal to 1 inch, and one division is equal to the 300dth part of an inch. 2. For measuring the magnifying power of other telescopes. This is done by measuring the diameter of the pencil of light at the eye-end of the telescope in question. For, if we divide the diameter of the object lens by the diameter of this pencil of light, the quotient will express how many times that telescope magnifies in diameter. Thus, suppose that 300 divisions of the micrometer are equal to the apparent extension of 1 inch—that the pencil of light is measured by 4 of these divisions—and that the diameter of the object lens measures 1 inch and 2 tenths:—Multiply 1.2 by 300, and the product 360, divided by 4, gives 90 for the magnifying power of the telescope.

Problems which may be solved by this micrometer. I. The angle—not exceeding one degree—which is subtended by an extension of 1 foot, being given, to find its distance from the place of observation:—Rule 1. If the angle be expressed in minutes, say, as the given angle : is to 60 :: so is 687.55 : to a fourth proportional, which gives the answer in inches. 2. If the angle be expressed in seconds, say, As the given angle : is to 3600 :: so is 687.55 to a fourth proportional, which expresses the answer in inches. 3. If the angle be expressed in minutes and seconds, turn it all into seconds, and proceed as above. Example, at what distance is a globe of 1 foot in diameter, when it subtends an angle of 2 seconds? 2 : 3600 :: 687.55 : (3600 × 687.55)/2 = 1237596 inches, or 103132½ feet = the answer required. II. The angle which is subtended by any known extension being given, to find its distance from the place of observation. Rule, Proceed as if the extension were of one foot, by Problem I, and call the answer B; then if the extension in question be expressed in inches, say, as 12 inches : are to that extension :: so is B : to a fourth proportional, which is the answer in inches. But if the extension in question be expressed in feet, then we need only multiply it by B, and the product is the answer in inches.—Example, At what distance is a man 6 feet high, when he appears to subtend an angle of 30´´? By Problem I, if the man were 1 foot high, the distance would be 82506 inches; but as he is 6 feet high, therefore multiply 82506 by 6, and the product is the required distance, namely 495036 inches, or 41253 feet.

For greater conveniency, especially in travelling, when one has not the opportunity of making such calculations, the following two tables have been calculated; the first of which shows the distance answering to any angle from one minute to one degree, which is subtended by a man whose height is considered an extension of 6 feet, because at a mean, such is the height of a man when dressed with hat and shoes on. These tables may be transcribed on a card, and may be kept always ready with a pocket telescope furnished with a micrometer. Their use is to ascertain distances without any calculations; and they are calculated only to minutes, because with a pocket telescope and micrometer, it is not possible to measure an angle more accurately than to a minute. Thus, if we want to measure the extension of a street, let a foot ruler be placed at the end of the street; measure the angular appearance of it, which suppose to be 36´, and in the table we have the required distance against 36´, which is 95½ feet. Thus also a man who appears to be 49´ high, is at the distance of 421 feet. Again, Suppose the trunk of a tree which is known to be 3 feet in diameter be observed to subtend an angle of 9´½. Take the number answering to 9´ out of the table, namely 382, and subtract from it a proportional part for the half minute, namely 19.1, which subtracted from 382, leaves 362.9. This multiplied by 3, the diameter of the tree, produces 1087.7 feet = the distance from the object end of the telescope.

Angles subtended by an extension of one foot at different distances.				Angles subtended by an extension of six feet at different distances.
Angles Minutes.	Distances in feet.	Angles Minutes.	Distances in feet.	Angles Minutes.	Distances in feet.	Angles Minutes.	Distances in feet.
1	3438	31	110.9	1	20626.8	31	665.4
2	1719	32	107.4	2	10313.	32	644.5
3	1146	33	104.2	3	6875.4	33	625.
4	859.4	34	101.1	4	5156.5	34	606.6
5	687.5	35	98.2	5	4125.2	35	589.3
6	572.9	36	95.5	6	3437.7	36	572.9
7	491.1	37	92.9	7	2946.6	37	557.5
8	429.7	38	90.4	8	2578.2	38	542.8
9	382	39	88.1	9	2291.8	39	528.9
10	343.7	40	85.9	10	2062.6	40	515.6
11	312.5	41	83.8	11	1875.2	41	503.1
12	286.5	42	81.8	12	1718.8	42	491.1
13	264.4	43	79.9	13	1586.7	43	479.7
14	245.5	44	78.1	14	1473.3	44	468.8
15	229.2	45	76.4	15	1375.	45	458.4
16	214.8	46	74.7	16	1298.1	46	448.4
17	202.2	47	73.1	17	1213.3	47	438.9
18	191	48	71.6	18	1145.9	48	429.7
19	181	49	70.1	19	1085.6	49	421.
20	171.8	50	68.7	20	1031.4	50	412.5
21	162.7	51	67.4	21	982.2	51	404.4
22	156.2	52	66.1	22	937.6	52	396.7
23	149.4	53	64.8	23	896.8	53	389.2
24	143.2	54	63.6	24	859.4	54	381.9
25	137.5	55	62.5	25	825.	55	375.
26	132.2	56	61.4	26	793.3	56	368.3
27	127.3	57	60.3	27	763.9	57	361.9
28	122.7	58	59.1	28	736.6	58	355.6
29	118.5	59	58.2	29	711.3	59	349.6
30	114.6	60	57.3	30	687.5	60	343.7

In this way the distance of a considerably remote object, as a town or building at 10 or 12 miles distant, may be very nearly determined; provided we have the lineal dimensions of a house or other object that stands at right angles to the line of vision. The breadth of a river, of an arm of the sea, or the distance of a light house, whose elevation above the sea or any other point, is known, may likewise in this manner be easily determined.