Coordinate

“ In astronomy, the apparent places of the stars are reduced to order by the use of a system of astronomical coördinates, just as we reduce to order our knowledge of geographical positions on the earth’s surface by the conceptions of latitude and longitude. ”

“ One can also find in the same way the surface area of a solid of revolution.
Example. The curve is revolving about the axis of . Find the area of the surface generated by the curve between and .
A point on the curve, the ordinate of which is , describes a circumference of length , and a narrow belt of the surface, of width , corresponding to this point, has for area . The total area is
.
Areas in Polar Coordinates.
When the equation of the boundary of an area is given as a function of the distance of a point of it from a fixed point (see Fig. 61) called the pole, and
Fig. ”

“ An axis is any definite direction in space. We may suppose it defined in Cartesian coordinates by its three direction-cosines l, m, n, or, better still, we may suppose a mark made on the surface of a sphere where the radius drawn from the centre in the direction of the axis meets the surface. ”

“ VI. A map drawn by this rule will have the following properties:
1. The intersections of the meridians and parallels will be rectangular.
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2. The distances north and south will be exact; and any meridian will serve as a scale.
3. The parallels thro’ z and y, where the line Rr cuts the arc Ll, or any small distances of places that lie in those parallels, will be of their just quantity. At the extreme latitudes they will exceed, and in mean latitudes, from x towards z or y, they will fall short of it. ”

“ See also BACK RANGE, LEADING LINE, MAGNETIC RANGE, MULTIPLE RANGES. 2. Distance in a single direction or along a great circle. 3. The extreme distance at which an object or light can be seen is called VISUAL RANGE. When the extreme distance is limited by the curvature of the earth and the heights of the object and the observer, this is called geographic range; when the range of a light is limited only by its intensity, clearness of the atmosphere, and sensitiveness of the observer’s eyes, it is called luminous range. 4. The extreme distance at which a signal can be detected or used. ”

“ But we have chosen our example of a greater breadth than can often be required, on purpose [558] to shew how high the errors can ever arise; and how they may, if it is thought needful, be nearly estimated and corrected.
Write down, in a vacant space at the bottom of the map, a table of the errors of equidistant parallels, as from five degrees to five degrees of the whole latitude; and having taken the mean errors, and diminished them in the ratio of radius to the sine of the mean inclination of the line of distance to the meridian, you shall find the correction required ”

“ We imagine that the particle is located at the origin of the coordinates of the "unprimed" system (x, y, z, t) and has the velocity components with respect to this system, and ask for the equations of motion. This question can be answered unequivocally by the fact that we first think of the mass points as the origin of a new reference frame, which moves against the original frame with the constant velocity components . The x-axis of this system may coincide with the direction of the velocity q of the mass point, whose size is expressed by 3) . ”

“ For orientation in three dimensions of space we must extend our ideas and consider position with reference to east and west and up and down. A man lying on his side with his head to the north and looking east, would not be similarly oriented to a man standing upright and looking east. To provide for the complete specification of how a body is oriented in space we must have in the body a plane of reference, and a line of reference in this plane, belonging to the body and moving with it. ”

“ When you wish to give a general outline of size and configuration, you should devise for yourself some rule which may be adhered to more or less. After having laid down that the breadth of the space occupied by the mountains which run in a direction due east, as well as by the sea which reaches to the Pillars of Hercules, is 3000 stadia, would you pretend to estimate different lines, which you may draw within the breadth of that space, as one and the same line? ”

“ Any definite line is one along which remains constant while varies continuously, and inversely any definite line is one along which remains constant while varies continuously. In much the same way, on the earth’s surface a latitude line or parallel is one along which the latitude remains constant and the longitude varies, whereas a longitude line or meridian is one along which the reverse is true. ”

“ Latitude and longitude, which lie at the basis of all descriptive geography, are determined by observation. No map deserves the name, on which the position of important points has not been astronomically determined. Some even of our most important political and administrative arrangements depend upon the coöperation of this science. Among these I may mention the land system of the United States, and the determination of the boundaries of the country. ”

“ If we turn round and go in the opposite sense, then the area will be to the left if it was first to the right, and vice versa. If we give the boundary a definite sense, and go round in this sense, then the area is said to be either of the one or of the other sense according as the area is to the right or to the left. ”

“ The latitude, which you have found, measures your distance north or south from the equator or the pole. To find your longitude, you want to find your distance east or west from the meridian of Greenwich. Now if any one would build a good tall tower at Greenwich, straight into the sky,—say a hundred miles into the sky,—of course if you and I were east or west of it, and could see it, we could tell how far east or west we were by measuring the apparent height of the tower above our horizon. ”

“ I mean that every Point in you—for you are a Square and will serve the purpose of my illustration—every Point in you, that is to say in what you call your inside, is to pass upwards through Space in such a way that no Point shall pass through the position previously occupied by any other Point ”

“ If you should travel round the earth from east to west, you would go in a ring or circle. If you went round it from north to south, and from south to north again, you would also go in a circle.
Such circles are not really drawn on the surface of the earth. We only imagine them, and they are called imaginary circles. They are represented by the lines that you see on the map of the hemispheres, and on other maps. These lines are drawn from north to south, and from east to west. Those drawn from north to south are called meridians. Those from east to west are called parallels. ”

“ On their side, geometers, when measuring the size of the entire earth, avail themselves of the data furnished by the natural philosopher and astronomer; and the geographer on his part makes use of those of the geometer.
3. The heavens and the earth must be supposed to be divided each into five zones, and the celestial zones to possess the same names as those below. The motives for such a division into zones we have already detailed. These zones may be distinguished by circles drawn parallel to the equator, on either side of it. ”

“ A circle may be described with any given point as a center and any given line as a radius. This involves the use of the second of the two geometric instruments, the compasses.
5. Any figure may be moved from one place to another without altering the size or shape. This is the postulate of the homogeneity of space, and asserts that space is such that we may move a figure as we please without deformation of any kind. It is the basis of all cases of superposition. ”

“ Assuming the earth to be spheroidal, let A and B be two stations on its surface, and let the latitude and longitude of A be known, also the azimuth of B at A, and the distance between A and B at the mean sea-level ”

“ The position of a straight line, or detail, on the map will be correct when its actual and plotted position on the ground and map makes equal angles with another known line and intersects it in a known point, the position of which line and point on the ground has been previously ascertained to be correctly represented on the map. ”

“ If, in a certain diagram, whose geometrical history we know, we want to test whether two Lines, passing through a common point, have, or have not, 'the same direction,' we have simply to take any other point on one of the Lines, and observe whether the other Line does, or does not, pass through it. ”

“ In any circle, the nearer the points of the circumference are to the centre, the closer they will necessarily be to one another. As we draw nearer, each for himself, to our Centre, we shall feel that we have approximated to all those who stand round the same centre, and draw from it the same life. ”

“ Now, were the earth a perfect sphere, the meridian of the earth would be a perfect circle, and the distance between any two places, differing one degree in latitude, would be exactly equal to the distance between any other two places, differing in latitude to the same amount. ”

“ Now unroll the cylinder and the projection will appear as in fig. 2. The whole world is now represented as a rectangle, each parallel is a straight line, and its total length is the same as that of the equator, the distance of each parallel from the equator is sin l (where l is the latitude and the radius of the model earth is taken as unity) . The meridians are parallel straight lines spaced at equal distances.
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This projection possesses an important property. ”

“ Study a good map—a quadrangle of the U. S. Geological Survey, for instance. Note what things are represented, and how representation is made: study the map, until it is thoroughly understood.
There are three factors with which the map-maker deals: direction, distance, and elevation. With the first, he must always reckon, and usually with the second and the third as well.
Direction is fundamental. Suppose there are three dominant points in the area to be mapped, relatively situated as here indicated. ”

“ An arc of the meridian may be measured by observing the latitude of its extreme points, and then measuring the distance between them in feet or fathoms; the distance thus determined on the surface of the earth, divided by the degrees and parts of a degree contained in the difference of the latitudes, will give the exact length of one degree, the difference of the latitudes being the angle contained between the verticals at the extremities of the arc. ”

“ For instance, a series of projections may be constructed for zones, each having a range of 10° of latitude, from the equator to the pole. Treating the earth as a sphere and using the above formulae, the series will possess the following properties: the meridians will all be true to scale, the area of each zone will be correct, the scale errors of the limiting parallels will all be the same, so that the length of the upper parallel of any zone will be equal to that of the lower parallel of the zone above it. ”

“ In order that directions on the map and on the ground shall correspond, it is necessary for the map to be oriented, that is, the true meridian of the map must lie in the same direction as the true meridian through the observer's position on the ground, which is only another way of saying that the lines that run north and south on the map must run in the same direction as the lines north and south on the ground. ”

“ Such is the conception of an accurate map which is an attempt to produce on a plain surface or sheet of paper, a horizontal projection of objects on the ground, which will show the relative positions of every detail on any desired scale with as little distortion as possible, and on which distances may be measured in any direction, and areas computed with a degree of accuracy only limited by the scale. ”

“ Of what value will it be to him to know that certain lines indicate a mountain range or river, unless he has an approximate idea of what a mountain range or river is?
For the use of the more advanced pupils of the higher grades, who see the relation of structural environment to man in his development as a nation, the relation of natural structural divisions to political divisions, these maps should have lines drawn upon them to indicate the boundaries of such divisions. ”

“ These measures of arcs of the meridian, as measurements of a known and selected portion of the surface of the spheroidal globe of the earth, have, more or less, fixed mathematical relations with the axis of the earth; as the circumference of a sphere has an exact mathematical ratio to its diameter. ”

“ The horizontal map distances should be related to the other horizontal distances of the map, and the altitudes to other altitudes, and these with reference, also, to the tabulated lists found in every geography, of the heights of mountain peaks and lengths of rivers.
“All knowledge of external things comes through observation, comparison, and judgment.” To judge of great altitudes, one must have a knowledge of the heights within experience. ”

“ The co-ordinates of the principal stations of a trigonometrical survey are usually the spherical co-ordinates of latitude and longitude; those of a traverse survey are always rectangular, plane {or a small area but spherical for a large one. ”

“ The position of a point in space is intelligently measured only when we know that three definite and independent co-ordinates are required to locate the point and have determined the numerical values of these three co-ordinates. ”

“ The habit of locating the events recorded, of tracing upon the map the route of an army, or the line of an important road or canal, and observing the impediments or natural obstructions to be overcome, with the great advantages to be derived therefrom, together with inferences as to the time and labor required, has the effect of making the study of history of living interest, especially if the map used indicates such surface structure. ”