In south latitudes the azimuth is found in the same manner; only, the sun's azimuth is found from the S.

Then, to find the variation, place your needle below the centre of the quadrant, set it level, and find how many degrees the pole, or object, in the sun's azimuth, bears from the north by the needle; and the difference between that and the azimuth found by calculation, is the variation of the needle sought.

If the sun ascends, or descends with little obliquity, a meridian line may be fixed pretty exactly this way, because a small inaccuracy in the altitude of the sun's centre, will not be sensible in the azi, muth. But, when the sun does not rise high on the meridian, this method is not to be relied on, when great exactness is necessary; for then every inaccuracy in latitude, altitude, and refraction, occasions severally a greater error in the azimuth, To mark the meridian line on the ground, place the centre of a theodolite, or Hadley's quadrant, where the centre of the quadrant was when the sun's altitude was taken, and putting the index to the degree and minute of the azimuth, direct, by waving your hat towards one side or the other, a pole to be set up, making an angle with the former pole placed in the azimuth, equal to the sun's azimuth found; and that last-placed pole will be in the meridian, seen from the centre of the quadrant.

N. B. The sun's declination in the tables must be corrected by the variation arising from the difference of the time between your meridian and that of the tables; and also for the variation of declination for the hours before, or after noon, at which the sun's altitude was taken.

The two last problems are constructed and fully explained in Robertson's, Mackay's, and other

Treatises on Navigation; it is, therefore, needless to be more particular here.

A COURSE

OF

PRACTICAL GEOMETRY*

ON THE GROUND,

BY ISAAC LANDMAN,

PROFESSOR OF FORTIFICATION AND ARTILLERY,
ROYAL MILITARY ACADEMY AT WOOLWICH,

It is as easy to trace geometrical figures on the ground, as to describe them on paper; there is, however, some small difference in the mode of operation, because the instruments are different. A rod, or chain, is here used instead of a scale; the spade, instead of a pencil; a cord fastened to two staves, and stretched between them, instead of a rule, the same cord, by fixing one of the staves in the ground, and keeping the other moveable, answers the purpose of a pair of compasses; and with these few instruments every geometrical figure necessary in practice may be easily traced on the ground.

PROBLEM 1. To draw upon the ground a straight line through two given points A, B, fig. 1, plate 27. Plant a picket, or staff, at each of the given

* I have been obliged to omit in this Course, so liberally communicated by Mr. Landman, the calculations that illustrate the examples; this, I hope, will not in the least lessen its use, as this work will fall into the hands of very few who are ignorant of the nature and application of logarithms.

points A B, then fix another, C, between them, in such manner, that when the eye is placed so as to see the edge of the staff A, it may coincide with the edges of the staves B and C.

The line may be prolonged by taking out the staff A, and planting it at D, in the direction of B and C, and so on to any required length. The accuracy of this operation depends greatly upon fixing the staves upright, and not letting the eye be too near the staff, from whence the observation is made.

PROBLEM 2. To measure a straight line.

We have already observed, that there is no operation more difficult than that of measuring a straight line accurately; when the line is short, it is generally measured with a ten-feet rod; for this purpose, let two men be furnished each with a ten feet rod, let the first man lay his rod down on the line, but not take it up till the second has laid his down on the line, so that the end may exactly coincide with that of the first rod; now let the first man lift up his, and count one, and then lay it down at the end of the second rod; the second man is now to lift up his, and count two; and thus continue till the whole line is measured. Staves should be placed in a line at proper distances from each other by Problem 1, to prevent the operators from going out of the given line.

When the line is very long, a chain is generally used; the manner of using the chain has been already described, page 194.

PROBLEM 3. To measure distances by pacing, and to make a scale of paces, which shall agree with another, containing fathoms, yards, or feet.

In military concerns, it is often necessary to take plans, form maps, or procure the sketch of a field of battle, and villages of cantonment, or to reconnoitre fortified places, where great accuracy is not required, or where circumstances will not allow the

use of instruments; in this case it is necessary to be well accustomed to measuring by the common pace, which is easily effected by a little practice; to this end measure on the ground 300 feet, and as the common pace is 24 feet, or 120 paces in 300 feet, walk over the measured space till you can finish it in 120 paces, within a pace or two,

Example. Let us suppose that we have the map of a country containing the principal objects, as the villages, towns, and rivers, and it be necessary to finish it more minutely, by laying down the roads, single houses, hills, rocks, marches, &c. by measuring with the common pace; take the scale belonging to the map, and make another relative to it, in the following manner, whose parts are paces. Let the scale of the map be 200 fathoms, draw a line AB, fig. 3, plate 27, equal to this scale, and divide it into four equal parts, AC, CD, &c. each of which will represent 50 fathoms; bisect AC at E, divide A E into five equal parts, A e, ef, fg, &c. each of which will be five fathoms; draw G H parallel to AB, and at any distance therefrom then through the points of division A, e f, g h, E, C, &c. draw lines perpendicular to AB, and cutting GH, which will be thereby divided into as many equal parts as A B. Two hundred fathoms, at 24 feet per pace, is 480 paces; therefore write at H, the last division of G H, 480 paces, at I 360, at K 240, and so on. To lay down on the plan any distance measured in paces, take the number of fathoms from the line A B, corresponding to the, number of paces in GH, which will be the distance to be laid down on the plan.

When plans, or maps, are taken by the plain table, or surveying compass, this is an expeditious method of throwing in the detail, and a little prac, tice soon renders it easy.

;

A TABLE,

For reducing the common pace of 2 feet into feet and inches

Paces Feet Inc. P. Feet Inc. P Feet Inc. P. Feet Inc. P. Feet Inc.

Distances of a certain extent may be measured by the time employed in pacing them; to do this, a person must accustom himself to pace a given extent in a given time, as 600 paces in five minutes, or 120 in one minute: being perfect in this exercise, let it be required to know how many paces it is from one place to another, which took up in pacing an hour and a quarter, or 75 minutes. Multiply 75 by 120, and you obtain 120, the measure required.

This method is very useful in military operations; as for instance, when it is required to know the itinerary of a country for the march of an army, or to find the extent of a field of battle, an encampment, &c. It may be performed very well on horseback, having first exercised the horse so as to make him pace a given space in a determinate time, and this may be effected in more or less time, according as he is trained, to walk, trot, or gallop the original space. PROBLEM 4. To walk in a straight line from a proposed point to a given object, fig. 2, plate 27.