PHILOSOPHICAL PAPERS. " PHILOSOPHICAL USES of a COMMON WATCH, by the REVEREND WILLIAM PEARSON. [From the third Volume of NICHOLSON'S JOURNAL of NATURAL PHILOSOPHY, CHEMISTRY, and the ARTS.] HE theoretic philosopher, philosophical uses that may be made are many observations and experiments in different departments of science, the accuracy of which depend greatly, and some of them entirely, on the accurate measurement of minute portions of time; such, for instance, as the determination of the velocity of sound, the nature of the descent of falling bodies, the measure of the sun's diameter, the distance of two contiguous, or at least apparently contiguous, heavenly bodies taken at their passage over the meridian, and the distance of places from the difference of the velocity of light and sound. A pendulum to swing seconds has usually been applied for these and similar purposes, and in an observatory is found to be very convenient; but a watch, by being more portable, is calculated to be more general in its application, and will measure smaller portions of time than any other instrument that has been invented* ; besides, it possesses this peculiar advantage, that in all gated fields of science, may gather the sweets as he passes along, and grati. fy his taste with fruits that he has had no share in rearing, at a very inconsiderable expense; but the pleasure which the practical man of science derives from the results of his successful experiments is much more exquisite: the previous impression made by contemplating the harmony that appears in the laws of nature; the satisfaction of proving the agreement or discrepancy between theory and practice; the prospect of benefiting society in some shape, and, perhaps, also the gratification of a certain degree of vanity, all concur in stimulating his exertions. If he should labour under any inconvenience in procuring the best means of prosecuting his labours, he will naturally avail himself of the best substitutes that the circumstances of his situation will allow. From this consideration it becomes a matter of surprise that a more general attention is not paid to the "The beat of a watch is quicker than of any other chronometer in general use; but there have been instruments made to divide the second into 100 parts. One of these, made by Whitehurst, and regulated by a fly, repeatedly measured the time of fall of a leaden bullet (in some experiments which I saw) with no greater variation than one hupdredth part of a second.-N.". situations situations the beats thereof may be counted by the year, at the same time that the object of observation is viewed by the eye, so that no loss is incurred, as must inevitably happen, when the eye is used to view both the object and pendulum or second index in succession, though it be ever so quick.-But it will be objected here, no doubt, that few watches measure time accurately, and also that, from the different constructions of watches, the times corresponding to their beats vary in a very considerable degree. I allow these objections to be true, and conceive that the reason may be attributed to them, why the beat of a watch is not generally applied as the measure of the lowest denomination of subdivisions of time: I shall therefore endeavour in this paper to obviate these objections, by showing how any tolerably good watch, whatever be its construction, may be applied with advantage to many philosophical purposes. "We must, in the first place, consider that the portions of time which I propose to have measured by a watch are small portions only, and those to be counted not by a second hand, as is the custom with medical men, but altogether by the beats; in which case, if the watch be not liable to lose or gain time considerably in a day, the error in the rate of going will be extremely minute in the time corresponding to any number of beats that the memory can retain, or that the purposes to which I propose the application to be made will require: and even if the error in the rate of going be considerable, so as to a mount to many minutes in a day, as it is uniform, it may easily be allowed for by a correction. Hence the first objection, which relates to the error occasioned by the rate of going of any watch, will constitute no real obstacle to its application in the ascertaining of small portions, of time, provided a sudden change of temperature be avoided at the time of using it; for it will be necessary that the rate of going be estimated when the temperature is the same, or very nearly the same, as when the watch is used for philosophical purposes; so that if it is usually worn in the pocket, it may be held in the hand to the ear, but if it be hanging in a room or in the open air where the rate of going is ascertained,it must be hung near the ear, under similar circumstances, where any observation is intended to be made by it. "As to the other objection, which applies to the variation in the lengths of the beats of two different watches, owing to the difference of their constructions, though they indicate hours and minutes alike, it may be removed very readily. All common watches have the same number of wheels and pinions, which are known by the same names, and placed, no matter how variously, so as to act together without interruption; but all watches have not their corresponding wheels and pinions divided into the same number of teeth and spaces, and to this circumstance it is entirely owing that the beats of different watches differ from each other. As the rate of going of a watch is regulated by the lengthening or shortening of a spring, without any regard being paid to the numbers which compose the teeth of the wheels and pinions, a great latitude is allowable in the "If the error were five minutes per day, the allowance would be upwards of one three hundredth part.-N." calculation 3 calculation of those numbers; of the number of hours that the watch which the different makers avail themselves according as the numbers on the engines they use for cutting the teeth require: but whatever the numbers may be of which the wheelwork consists, if we divide double the product of all the wheels, from the centre wheel to the crown wheel inclusively, by the product of all the pinions with which they act, the quotient will be invariably the number of beats of the watch in question in one hour; and again, if we divide this quotient by 3600, the number of seconds in an hour, this latter quotient will be the number of beats in every second, which may be carried to any number of places in decimals, and be copied upon the watch-paper for inspection whenever it may be wanted. "When any particular watch is cleaned, the workman may be directed to count, and return in writing, the numbers of the centre wheel, the third wheel, the contrate wheel, and the crown (balance) wl cel, and also of the three pinions which they actuate, respectively, from which the calculation of the length of a beat is easily made by the rule just given, and when once made will apply in all instances where that individual watch is used. It will be remarked here, that no notice is taken of the wheels and pinions which constitute the dialwork, nor yet of the great wheel and pinion with which it acts: the use of the former of these is only to make the hour and minute hands revolve in their respective times, and may or may not be the same in all watches; and the use of the latter, the great wheel and its pinion, is to deterinine, in conjunction with the number of spirals on the fusee, shall continue to go, at one winding up of the chain round the barrel of the main-spring: all these wheels and pinions therefore, it will be perceived, are unnecessary to be taken into the account in calculating the beats per hour. The reason why double the product of the wheels specified is taken in the calculation is this, that one tooth of the crown wheel completely escapes the palats at every two beats or vibrations of the balance. A few examples will render the general rule perfectly intelligible. Let us take for the first example the numbers of a common watch given by Mr. Emerson in his "Tracts," which, according to his method of arrangement, stand thus, 48 great wheel, 6-48 third wheel "Now, omitting the great wheel and its pinion of 12, we have 54 x 48 × 48 × 15 × 2 =3732480 for double the product of the specified) wheels, and 6×6×6=216 for the product of the specified pinions; also 3732480 216 = 17280 are the number of beats in an hour, and 17280 3000 =4,8 the exact number of beats per second accordingly, Mr. Emerson says that this watch makes about ber of spirals on the fusce is 7; 4 beats in a second.' The numtherefore, 7x 48 12 25 is the num ber of hours that the watch will go at one winding up: likewise 'the the dial-work 40 36 1440 = 12 shows that whilst the first driving pinion of 10 goes twelve times round, the last wheel of 36 goes only once; whence the angular velocity of two hands carried by their hollow axles are to each other as 12 to 1. "For a second example, I with take a watch which is in my own possession, the numbers of which in the calculation of beats per second will be thus: 60 × 60 × 60 × 13x2=5616000, double the product of the wheels; and 8+8×6= 384 the product of the pinions; then 5616000 384 14625 will be the beats in an hour, and 4,0625, the beats per second. "Besides this I have examined two other common watches, one of which requires this calculation: 54 x 52 x 52 x 13 x2=3796416 for double the product of the wheels, and 6×6×6=216 for the product 3796416 of the pinions; therefore- 216 =17576 are the beats in an hour, and 17576 3600 =4,882, the beats per second by this watch also double the product of the wheels of the other, viz. 56×51×50×13×2 is 3712800, and the product of the pinions, as in the last, 6×6×6=216; conse3712800 quently 216 gives 17168 beats in an hour, which divided by 3600 gives 4,7746 for the beats per second. "These four examples, it is presumed, will render the method of ascertaining the beats per second in any watch sufficiently easy for any person who is acquainted with common arithmetic. "It remains now for an instance or two to be adduced for the application of the beats of a watch to philosophical purpo es, in order to shew the practical utility of the method here proposed of measuring very small portions of time. "Let us suppose, for one instance, with D. Herschel, that the annual parallax of the fixed stars may be ascertained by observing how the angle between two stars, very near to each other, varies in opposite parts of the year. For the purpose of determining an angle of this kind, where an accurate micrometer is wanting, let a telescope that has cross-wires be directed to the stars when passing the meridian, in such a manner that the upright wire may be perpendicular to the horizon, and let it remain unmoved as soon as the former of the two stars is just coming into the field of view; then fixing the eye to the te lescope and the watch to the ear repeat the word one along with every beat of the watch before the star is arrived at the perpendicular hair, until it is in conjunction with it, from which beat go on two, three, four, &c. putting down a finger of either hand at every twenty till the second star is seen in the same situ at the commencement of the countation that the leading one occupied ing; then these beats divided by the beats per secend, marked on the watch paper, will give the exact number of uncorrected seconds, by which the following star passes later over the meridian than the lead ing one: when these seconds and part of a second are ascertained, we have the following analogy for determining the angle, which includes also the correction, name. ly, as 23h. 56′ 4′′, 098, (the length of a sidereal rotation of - the earth), plus or minus the daily error in the rate of going: are to 360°:: so is the number of observ-, ed seconds of time: to the quantity of the horizontal angle required. The watch is here supposed to be regulated to show solar time; but if it should be regulated exactly for sidereal time, instead of 23h. 56′ 4".098 we must use exactly 24 hours in the analogy. "As a second instance, let it be required to ascertain the distance of the nearer of two electrified clouds from an observer, when there are successive peals of thunder to be heard a little time before the expected repetition of a flash of lightning place the watch at the ear, and commence the numbering of the beats at the instant the flash is seen, as before directed, and take care to cease with the beginning of the report; then the beats converted into seconds, with the proportional part of the daily error added or subtracted, will give the difference of time taken up by the motion of the light and sound: if, lastly, we suppose light to be instantaneous at small distances, the distance of the nearer cloud will be had by multiplying the distance that sound is known to pass through in a second by the number of observed seconds obtained from the beats that were counted. "Many more instances might be here pointed out, in which the beats of a good watch would be extremely serviceable in the practical branches of philosophy; but the occur rences of such instances will always point out the propriety of the application, when it is once known and practised. watch, which is, that it is free from any error that might arise from the graduations of a dial-plate, or nnequal divisions in the teeth of wheels and pinions, where the seconds are counted by a hand. "In order to introduce this method of measuring small portions of time accurately, it is desirable that a watch be constructed so as to make an exact number of beats per second without a fraction, for then the reduction of beats into seconds would be more readily made. With a view of promoting this object, I have calculated numbers for a watch, which will produce the desired effect, and which, as they are equally practicable as those in use, I shall insert in this paper, in hopes of hearing at some future period that they have been adopted by some good workman. By the method of arrangement already given, the numbers proper for such a watch, as will indicate hours, minutes, and seconds, by three hands, and also make just four beats per second, will stand thus, viz. 50 great wheel 10-60 centre wheel 8-64 third wheel 8-48 contrate wheel Dial-work as usual. for ascertaining the beats per second "By the preceding general rule in any watch, the calculation of 64 x 48 x 15 x2=5529600, and 8 these numbers will be thus: 60 x x 8 x 6384; then "I shall therefore only mention one further advantage which seems peculiar to this mode of counting a limited number of seconds by a 3600 5520600 384 14400, the beats in an hour, and 14400 4 exactly, for the beats per second; |