Front Page Titles (by Subject) APPENDIX C.: VELOCIMETER. An Instrument for Calculating Velocities on Railways, &c. - An Autobiography, vol. 1
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APPENDIX C.: VELOCIMETER. An Instrument for Calculating Velocities on Railways, &c. - Herbert Spencer, An Autobiography, vol. 1 
An Autobiography by Herbert Spencer. Illustrated in Two Volumes. Vol. I (New York: D. Appleton and Company 1904).
Part of: An Autobiography
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[From the “Civil Engineer and Architect’s Journal” for July, 1842.]
The instrument represented in the annexed plate, which I have named a “Velocimeter,” is intended to supersede the long calculations, frequently necessary, in obtaining velocities in engine trials.
When the times of passing the quarter mile posts only are noted, such an apparatus is hardly called for, since, the distances being constant, a table may readily be made out which will give the velocities due to the different times; but it is a common practice, and perhaps a more satisfactory one, to note the times taken in traversing the several gradients, where the distances as well as the times are variable. The lengths of the inclines are generally fractional, and probably no two are the same, and none of the times of travelling over them are equal; consequently each case involves a distinct calculation, and where the trials have been extensive, several days may be occupied in making these reductions. It is, therefore, a desideratum to have some other means of obtaining the velocities, than that afforded by the ordinary methods of calculation.
The instrument devised for this purpose, is another application of that very important geometrical principle—the equality of the ratios of the sides of similar triangles. In the right angled triangle ABC (fig. 1), let AB be taken to represent any given number of minutes and seconds, and AC the number of miles and chains passed over in that time. Then, if AB be produced until it becomes equivalent to an hour, and from its extremity D, a perpendicular be drawn intersecting AC produced in E, AE will represent the number of miles that would have been traversed in the hour had the motion been continued, that is, it will indicate the rate per hour at which the distance AC was travelled. Now, if AE be made to revolve round A, and to take any other positions, as AE′ or AE″, it is clear that the relations will still be the same, and that if any distances AC′, or AC″, be described in the time AB, AE′ and AE″ will indicate the respective rates per hour. If, in addition to this, BC be made moveable along AD, or, what is the same thing, if AD be divided into minutes and seconds, and lines be drawn from the divisions parallel to BC, we shall be able to adjust the revolving line, to any distances and times, within the limits that may be allowed by the arrangement.
It will probably be objected, that if the line AD, representing an hour, is to be divided into minutes and seconds, its length must be so great as to make the instrument too unwieldy for common use. This difficulty is, however, very readily surmounted.
If AD (fig. 2) be taken to represent a quarter of an hour, instead of an hour as in the last figure, it follows, that other things being the same, AE will represent one-fourth of the number of miles per hour; that is, if AE had four times the number of divisions, it would indicate the rate per hour; if, therefore, AE have two scales, one for adjustment and the other with divisions one-fourth the size for indication, the velocities may be read off as before. Or if it be desirable to make use of one-tenth of an hour, instead of one-fourth, we have only to make the indicating divisions one-tenth of the size of the adjusting divisions, and the same result will follow.
In the application of this principle to practice, the following arrangements are made:—AD is the scale of time, embracing in this case one-tenth of an hour, or six minutes; each minute includes 15 divisions, one of which will, therefore, represent 4 seconds, and as each of these may be readily bisected by the eye, the scale may be considered as divided into periods of two seconds each. AE is the scale of distance, turning on the centre A, the adjusting scale being divided into 4 miles, and each of these subdivided into 80 chains; the same space is divided on the indicating scale into 40 miles, and each of these into eighths, ten miles on the one scale being equivalent to one on the other, in consequence of the time scale extending only to one-tenth of an hour.
To obtain results by this apparatus, the revolving scale is moved until the division answering to the number of miles and chains passed over, is made to coincide with the division, representing the number of minutes and seconds, occupied in the transit; and this adjustment being made, the rate per hour is read off on the indicating scale, at its point of intersection with the line DB. For instance, a gradient 1 mile 25 chains long, is traversed in 2 minutes 48 seconds; what is the velocity? The divisions corresponding to these data being made to coincide, as shown at (a), the point of intersection on the indicating scale is examined, and the velocity found to be rather more than 28 miles per hour, which is the result given by calculation.
Again, a locomotive travels 1 mile 54 chains, in 4 minutes 40 seconds; required, the rate per hour. The revolving scale is moved as before, until the distance division 1 mile 54 chains at (b), is brought to (b′) on the division of 4 miles 40 seconds; the edge of the scale will then occupy the line A c′, and the point (c) on the scale will have arrived at the point of intersection (c′), showing the velocity to be rather more than 21½ miles per hour.
Of the three data time, distance, and velocity, any two being given, the third may be found, so that the apparatus may be employed in finding times, and distances, as well as velocities. Thus, having fixed the velocity at which the trains on a railway are to travel, and knowing the distances between the stations, the times of arrival may be ascertained, by adjusting the revolving scale to the required velocity, and noting the times corresponding to the given distances, and should the results be unsuitable, other velocities may be assumed, until the desired ends are fulfilled.
I have constructed two of these instruments, one for small, and the other for greater distances. The first (as far as I can remember, for it is not now in my possession), is about half as large again as the accompanying drawing, and has the same arrangement, except that the indicating scale extends to 45, instead of 40 miles, and the time scale has double the number of divisions, so that differences of a second are appreciable. The other has a time scale extending to 15 minutes, each minute being subdivided into periods of 4 seconds, so that differences in time of 2 seconds are available. The scale of distance has the adjusting scale divided into 11¼ miles, and each mile is subdivided into distances of 2 chains; the indicating scale extends to 45 miles, and each mile is divided into tenths. In both cases, the subdivisions of the time scale are made by lines of different colours, so as to avoid confusion.
These instruments, although made of Bristol board, and having a needle for the pivot of the revolving scale, gave results within one-eighth of a mile per hour of the truth; an approximation quite near enough for ordinary purposes. They were used for some time in engine trials, on the Birmingham and Gloucester Railway, and were found to answer very satisfactorily.
Derby, May 13, 1842.