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and coerced as best can be done-into whose probable error, resulting from the a plane surface, tangent to the earth at inaccuracy of the method employed, will the center of population, the various be less than the other error arising from normals, now normal to the same plane, the crude but inevitable custom of handbecome parallel lines, and the center of ling the population by communities inpopulation becomes a center of parallel stead of by individuals. Any greater forces. precision than this is not worth the imThus simplified, it is not a difficult mensely greater labor that it would cost. task to determine this center. But there It is now the object of this article to prois a flaw in this process. It results from pound some way, sufficiently rapid to be the impossibility of representing a spher- practicable, of attaining this degree of oidal surface upon a plane and preserv- exactness. ing all areas, proportions and distances As has been said, the spherical shell in their true values; in other words, the must be considered as a plane surface. surface of our spheroid is not capable of But there are numerous ways of doing development. When it is remembered this. Every style of map projection is a that the leverages which enter as factors transformation of this kind, and, unforinto the various moments are distances tunately, every one involves some disupon the earth's surface, it will be seen placement, distortion, or expansion or that a true expression of them is neces- contraction of areas and distances. We sary for a rigidly accurate determination now seek a projection which will incur a of the center of population. Such an minimum amount of error in dimensions expression can only be found by a com- and relative positions, and whose condiputation of the lengths of the arcs which tions will thus be most favorable for the constitute these lever arms. work at hand.

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Since the center of population is that As the leverages are to be the perpenpoint about which there is an equilibrium dicular distances from the local centers of moments, there must also be equili- to the great circle axes, it is important brium with respect to any two rectangu- that these great circles should appear on lar axes which may be drawn there- the map as straight lines, intersecting at through. Upon the earth's surface these right angles at the assumed center; or, rectangular axes become two great circles failing in that, that they should approach perpendicular to each other. It is ob- as nearly as possible to rectilinearity. viously most convenient to adopt for Cassini's projection is the only one this purpose the meridian through the known to the writer in which these two assumed center, and its perpendicular, great circles would be projected into which is the prime vertical at that point. right lines; but this method is attended It would now be necessary, in the method by so many disadvantages-among others under contemplation, to compute the the obliteration of our convenient division leverage of every group of population of a country into square degrees by its with respect to each of these axes. This reticulation of meridians and parallelsleverage is the shortest distance from that it may be dismissed from further the group to the line aforesaid; that is, consideration. an arc of a great circle perpendicular to In the polyconic projection, taking the the axis. So laborious a process as this, middle meridian through the assumed involving so many computations in center and making it one of the axes, it will geodesy and spherical trigonometry, is be projected into a right line, while its manifestly impracticable and quite out of co-ordinate axis-the prime vertical—will the question, and is mentioned here only not depart far from the same auspicious because of the theoretical interest it may condition; its greatest deviation from possess. rectilinearity within the limits of the

To return to the graphic and approxi- United States would not exceed forty mate mode, it is possible to construct a miles. This will not produce a curvamap of our country in such a manner, ture that will materially interfere with that the leverages taken by direct meas- the process of drawing perpendiculars to urement from the perpendiculars let fall it from the local population centers. The from the local population centers to the principal argument against the use of assumed axes, will give a national center the polyconic projection in this work is

Add to this the fact that the projection of the prime vertical through the intersection of middle meridian and parallel scarcely deviates from a straight line, its departure from rectilinearity within our boundary not exceeding ten miles, and the fitness of Bonne's projection for a population map is assured.

that uniformity of scale is not preserved, regard to the axes, and hence correct it being sacrificed in order to retain a relative positions, in which respect no natural angle of intersection between other projection offers equal advantages. meridians and parallels. Thus any two parallels of latitude on the map, instead of being concentric, and therefore equidistant throughout, as they are upon the earth's surface, separate more and more as they leave the middle meridian. With the middle meridian passing through Ohio, a projected degree of latitude in Oregon or Maine is much longer than in Michigan. In this way the population of those remote northwest and northeast regions will be given greater north leverages than they are entitled to, and the resulting tendency will be to draw the national center unduly northward. as possible to the true center, so that The preceding method is given for what it may be worth. It possesses at least the merit of convenience, as the data for the polyconic projection of a map are widespread in this country, and draughtsmen are familiar with their use. Moreover, with all of its faults, it is warranted to give results much more accurate than any hitherto attained.

Finally, we come to the consideration and adoption of Bonne's projection, which is by all odds the best adapted for delineating a portion of the earth's area, so that, having allotted weight to this map according to its density of population, the center of gravity of this plane will most accurately represent the center of population of the spherical surface.

Therefore, to find the center of population of the United States, first assume a center from an inspection of the map or from rough preliminary calculations. It is important that the assumed center should approximate in position as closely

the rectangular great circle axes through the former may be considered parallel, for the portion included within the United States, to the corresponding system through the latter, to which the first axes are to be shifted in the final adjustment. Make the assumed center the point of intersection of the middle meridian and parallel of a map projected after Bonne's method, and divided into square degrees. This map need not be drawn in detail. Outlines, boundaries, and the positions of those principal cities which are to be treated separately will suffice. Local centers of population may be taken from any ordinary map and transferred to this.

Draw through the assumed center the In Bonne's projection, as in the poly- projection of its prime vertical, tracing conic, the middle meridian becomes a its course by computing the latitudes of straight line. The central parallel, also its intersections with meridians which it as in the polyconic, is projected in the crosses. With this as one axis, and the arc of a circle whose radius is equal to rectilinear projection of the middle merithe slant height of a cone, tangent to the dian as its co-ordinate, proceed to find earth at this parallel, and of whose base the moments, with respect to these axes, this parallel is the circumference. All of the numerous collections of people other parallels are concentric with the into which it may be deemed advisable middle one, and are drawn at their to divide the country. Under the direcnatural distances apart. All lengths of tion of Mr. Henry Gannett, the very effiparallels are also correct. An equality of cient geographer of the Census Bureau, areas, upon the earth and on the map, is the population of the United States has thus preserved, although the shape of been grouped by square degrees, and, it those square degrees remote from the is to be hoped, will be published in that middle meridian and parallel becomes form. Such an arrangement will greatdistorted, and rhomboidal, their diagon ly facilitate all studies concerning the als being unequal, owing to the oblique geographical distribution of our people, intersection of meridians and parallels. and will be especially useful in finding This fault is of but little importance in the center of population. A determinathe present work, where the principal tion of the center by the use of square desideratum is uniformity of scale with degrees and the leading cities will re

quire about one thousand moments for population, by the old method, is erroneeach of the two axes. The leverages of ous to that extent. As an example, the these masses will be their perpendicular city of San Francisco is nearly a hundred distances, measured on the map, from miles south of the parallel aforesaid, and the axis in question. There is, to be has been given a south leverage of that sure, a slight element of error in this amount, when, in reality, it should have manner of finding the value of an arc of a north leverage of more than three a great circle, but it is so inconsiderable hundred miles. In the same manner, that it will not perceptibly affect the all other north and south moments are result, and it would be a thankless task invalidated by this fault of accepting a to attempt to eliminate it. small circle instead of a great circle for an axis.

After the summation of the moments thus found, and a comparison of the results, east with west, and north with south, the adjustment of the center to its true position is accomplished as described in the early part of this article.

The west leverage of San Francisco is also as much too large as its distance from the meridional axis by arc of parallel is longer than by an arc of great circle. But these excessive leverages in In determining the center of popula- the west are partly compensated for by a tion hitherto it has been customary to corresponding excess in the east, and the accept, for the rectangular axes through resulting error in the longitude of the the assumed center, the meridian and center is not so great; a mile or two parallel intersecting at that point; that would probably cover it. It is difficult is, a great circle and a small circle. to estimate, from a mere inspection of Leverages of the population masses the map, how great an error of latitude with regard to the parallel have been may affect the center of population, as measured along their respective meri- determined by the old method, but the dians, and are therefore arcs of great center, thus found, of the population of circles, as they should be; but the leverages about the meridional axis are the arcs of parallel intercepted between the local centers and that axis, and are, in comparison with the proper lever-arms, as the hypothenuse is to the perpendicular.

1880, falling in northern Kentucky, a few miles west of Covington, its true position is probably about thirty miles farther north and a trifle to the east; that is, in Butler County, Ohio.

It may seem improbable that the center should lie so near our northern boundary, but it should be remembered that our country, in general shape, is the segment of a zone, and if this segment were extended far enough in longitude its center of population would be situated entirely outside of the country, beyond its northern limit.

A moment's contemplation will show that this is equivalent to adopting, for the axes of a plane, a straight line and a curve intersecting at right angles, and then using curved instead of perpendicular lines for the leverages with respect to the rectilinear axis. Of these two erroneous processes, the latter will cause The difference between the plan herein the center a slight displacement in longitude, while the former will seriously affect its position in latitude, throwing it many miles too far southward.

proposed and the one hitherto followed, in determining the center of population, may be graphically illustrated by a comparison of Bonne's and Flamsteed's proAs heretofore determined, all north jections of a map. The latter is an leverages are deficient and all south exact counterpart of the plane surface leverages are excessive, except those ly into which the spherical shell is suping along the meridian of the center. posed to be projected in the old method; The greater the distance of a group of that is, its parallels occur at their natural population from this meridian, the intervals, but are straightened out into greater is the error of its moment. On right lines, all of which are perpendicuthe Pacific coast, for instance, the par- lar to the middle meridian, itself a allel of the center of population is more straight line. The middle parallel is than four hundred miles distant from taken as one of the rectangular axes, the prime vertical passing through that both of which, upon the map, are rectipoint. The leverage of any group of linear. Thus this process has a plausible

appearance, until it is remembered that cussions of the movement of our restless the distortion of this parallel into a right people. By computations made under line necessitates a great curvature in the his direction, and by independent deterprojection of the corresponding prime minations contributed by Prof. J. E. vertical, which is the real axis, and a Hilgard, of the Coast and Geodetic Survey, great displacement of areas. Flam- he was enabled to locate, upon his populasteed's projection, therefore, while it is tion charts, the successive positions of the easily comprehended by the popular center for each decennial period from the mind, which is accustomed to consider year 1790, when it was east of Baltimore, our country as a plane surface, across to 1870, when it had reached a point in which east and west courses run in Southern Ohio, forty-eight miles from straight lines, is really one of the most Cincinnati. For the details of its prountrue in existence. gress the reader is referred to the chap

In thus criticising past dealings with ter by General Walker, entitled "The this subject, the writer does not flatter Progress of the Nation," in his valuable himself that he has discovered an uncon- Statistical Atlas of the United States, scious use of faulty methods in the work and also to an article by Professor Hilof the able investigators who have pre- gard, published in Scribner's Monthly ceded him in the study of this matter. for June, 1872, under the heading of Taken at the best, its treatment must be "The Advance of Population in the approximate, and hitherto the newness United States." of the subject, an absence of popular interest in it, the pressure of more import ant work, and other exigencies of the public service, have combined to preclude any other than the most convenient and least laborious handling of this topic. It may even be doubted if, at the present day, it is advisable to bestow upon it the increase of labor entailed by the project announced in these pages; but in the future, as our methods of work become more and more refined, and especially as our northwest country and the Pacific Coast, remote from the center, become thickly populated, and the error of their influence upon the center is correspondingly increased, it will certainly be found necessary to attempt a closer approximation than has yet been accomplished.

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From these authorities it seems that the migration of the center of population, almost uniformly westward, has been at an average rate of fifty miles a decade, five miles a year, or seventy-two feet a day. This, then, is the measure of the progression of that imaginary, but important point, the "hub of the United States, which is destined to move, with slower and slower pace, until it shall reach, a century hence, the immediate valley of the Mississippi, where it will permanently settle. Though the West grow strong at the expense of its parent, the East, and the one flourishes as the other becomes effete, this shifting of power is no sudden and startling process, as will have been seen. The emigrant's ox-team, the type of all things slow, and by which this transfer of empire has been largely effected, outstrips à thousand fold the center of population

The old plan has an advantage of convenience in the fact that all distances along the meridians and parallels of our country, which are the leverages therein in its advance. used, are already computed and tabulated, in geodetic handbooks. And, although

the absolute situation of the center is not IT has been judicially decided in the accurately given by that method, it shows American law courts that a mail agent, correctly its relative positions, decade or traveling post-office servant, when after decade, and its constant westward traveling as such on railroad trains, is migration with its fluctuations to the not a passenger, and cannot therefore north and south. recover damages for injury in case of Through the enterprise of the Hon. accident; also, that a passenger riding Francis A. Walker, Superintendent of in a baggage car when there is room in the Census Bureau, the center of popu- the passenger cars cannot recover in lation was made an interesting feature of case of injury, if it shall appear that he the census reports for 1870, and will, would not have been injured if he had doubtless, find a place in all future dis- been in the passenger cars.

NEW MODE OF COMPENSATING BALANCES.

By CHARLES V. WOERD, Sup't Engineer of Waltham Watch Company.

THE form of compensation hitherto the spring is not certainly known; but employed in the balances of watches and sufficient is known in regard to it to chronometers can be made perfect only make it certain that the compensation for two critical temperatures, and is im- cannot be made perfect except at two perfect for intermediate or extreme temperatures. The expressions of the temperatures. Devices called secondary variations of these moments in respect or auxiliary compensation have been ap- to the temperature, give, when graphicplied to remedy this defect in a partial ally represented, two curved lines, and degree. The effect of the temperature the adjustment can be made only for

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is to modify the elastic force of the the two points where these curves inspring as well as to change its length, tersect.

and to change the form of the balance; The device consists in the substitution and perfect compensation requires that of an entirely different arrangement of the time of vibration of the balance shall bimetallic laminæ, in such a way that the be constant under all the vicissitudes of ratio of the moments of the balance and temperature to which the instrument of the spring shall remain constant durmay be subjected. This requires that ing all the temperatures to which the an arrangement of the parts shall be watch or chronometer may be subjected made such that the moment of inertia of while in use. the balance shall maintain a constant First, for the inner lamina steel is emratio to the moment of the elastic force ployed as usual, but the outer lamina is of the balance spring. The law of the constructed of a composition of higher change of the effective elastic force of expansibility than brass hitherto used

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