ia THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES FERGUSON'S LECTURES SELECT SUBJECTS, IN MECHANICS, | OPTICS, HYDROSTATICS, GEOGRAPHY, HYDRAULICS, ASTRONOMY, AND PNEUMATICS, DIALING. WITH NOTES AND AN APPENDIX, TO THE PRESENT STATE OF THE ARTS AND SCIENCES. By DAVID BREWSTER, A.M. The Second Edition. IN TWO VOLUMES, WITH A auAKTo VOLUME OF PLATES. Vohfrne IL EDINBURGH : TRIKTED FOR BELL Jj BRADFUTE, J. FAIRBAIRN ; MUNDELT., DOIG, $ STEvr.xsox, EDINBURGH; j. ij A. DUNCAN, GLASGOW ; AND T. OSTELL, AVE-MAHIA LANE, LONDON. 1806. -1O8UO , >U'J/,JK1 '! QIS1 CONTENTS of THE SECOND VOLUME. , -y/.v LECTURE X. On the principles and art of Dialing. . . . Dialing by the globe. . . . Dialing lines. . . . Tables of the sun's place and declination. . . . Tables of the equation of Time. . t . Rules for finding the latitude of places. Page 1 LECTURE XI. Of Dialing ..... Dialing by trigonometry ..... Babylonian and Italian dials. ... On the plac- ing f dials, and the regulation of time-keep* ers. . . 42 LECTURE XII. SJiewing how to calculate the mean time of any new or full moon, or eclipse, from the creation of the world, to the year of Christ 580O. .^ r/ Table of lunations. . . . Tables of the moon's mean motion from the sun, &c 71 173 494003 VI CONTENTS, SUPPLEMENT TO THE PRECEDING LECTURES BY THE AUTHOR. MECHANICS. Description of a new and safe crane with different poivers. 89 Description of a new and accurate pyrometer 94 On Barker's Mater-mill without wheel or trun- dle. 97 HYDROSTATICS. A machine for demonstrating that, on equal bot- : toms, the pressure ofjluids is in proportion to their perpendicular height, without any regard to their quantities. . 100 A machine to be substituted in place of the com- mon hydrostatic belloivs 104 The cause of reciprocating springs, and of ebbing andjlowing wells, explained. 106 HYDRAULICS. Account of Blakey'sjire engine. .... :! V -V 10 ^ Archimedes' s screw engine 113 Quadruple pump-mill for raising water. . . . 114 DIALING. Universal dialing cylinder. 118 Cylindrical dial .'. ^.ViV'? ..* :^. . . . 122 To make three sun dials on three different planes i^A .*&&.'i* WW..'-t-} & . MECHANICS, On the construction of undershot water wheels for turning machinery 1 3Q On the construction of the mill course 140 On the water wheel and its float-boards 1 47 On the spur wheel and trundle. . .... 154 fct ^TT A % ' On the formation, size, and velocity , o/* the millstone 1 58 On the performance of undershot mills 1 63 On the construction of new mill-wright' s tables 167 Explanation and use of the tables. . 176 Method of measuring the velocity of wa- ter 177 On horizontal mills 179 On double corn mills 184 On breast mills 1 88 Practical remarks on the performance and con- struction of overshot water wheels 1 92 On the method of computing the effective power of overshot wheels in turning machinery ib. On the performance of overshot and un- dershot mills 194 On the formation of the buckets, and the proper velocity of overshot wheels 196 EcsanCs underslwt wheel. 2O3 On Dr. Barker's mill. 205 Rules for its construction 2O8 CONTENTS. Account of an improvement in flour-mills. . 203 On the formation of the teeth of wheels, and the leaves of pinions .............. 21O On bevelled wheels, and the method of giving an epicycloidal form to their teeth ................ 23O On the formation of epicycloids, mechanic- ally, and on the disposition of the teeth on the wheePs circumference. . . . 234 On the formation of cycloids, and epicy- cloids, by means of points, and the me- thod of drawing lines parallel to them236 On the formation of the teeth of rack-work, the wipers of stampers, $c. . . . . ,, (V . : ^ . 241 On tJie nature and construction of windmills 258 Description of a windmill. ...... ib. On the form and position of windmill s.ails. .... ........ ,.,-> . 266 Tojind the momentum of friction. . 27O To find ike. velocity of the wind. . . 271 On the effect of windmill sails* ... 278 On horizontal windmills ....... 281 On wheel carriages. . . .."..'.' ^" v . ...... 295 On the formation of carriage wheels 296 On the position of the wheels. . . .'.312 On the line of traction, and the method by which horses exert their strength. 3 1 3 On the position of the centre of gravity, and the manner of disposing the load 317 On the thrashing machine. . . . , ..... 321 Oh thrashing machines driven by water323 On thrashing machines driven byhorses 327 On the power of thrashing machines. 332 On the nature of friction, and the method of di- minishing its effects in machinery ..... 334 On the nature and operation of fly wheels. * 353 On the construction and effect of machines. .* 61 Description of q simple aadpwverful capstone 381 Account of (in improvement up^n the balance S85 ^ mechanical method of. finding the centre of gravity. ................. 387 HYDRAULICS. .-\4\fls ViV :^.W\ORB nil \;i ftn^^rni^d 389 On the power of steaiy, engine^, and the method of computing it ...... 411 Description of a wqter-blvwvng machine. . . 415 Description of JVhitehurst' s machine for raising water by its momentum, and Montgoffier* s Hy- draulic ram .......... - ..... 419 OPTICS. On achromatic telescopes ........... 423 On achromatic object glasses, with tables of their radii of curvature ..... ib. On achromatic eye pieces ...... 444 On the construction of optical instruments, with tables of their apertures, $ c. and the method of grinding the lenses and mirrors of which they are composed ............... 452 f On the method of grinding and polishing lenses ............... ib. On the method of grinding and polishing the mirrors of reflecting telescopes. 457 On the single microscope ....... 462 On the double microscope ...... 467 On the refracting telescope ...... 468 On the Gregorian telescope ...... 472 On the Cassegrainian telescope. . . . 474 On the Newtonian telescope ..... 476 CONTENTS. Description of a new fluid microscope, invented by the editor. ... ..... .-; ; i 1 483 Account of an improvement on the camera ob- scum, and of a new portable one upon a large scale 48$ DIALING, Description of an analemmatic dial which sets itself. . . . ^i*.^: . . 489 Description of a new dial, invented by Lambert 496 ASTRONOMY. On the cause of the tides , , 498 LECTURES ON SELECT SUBJECTS. LECTURE X. THE PRINCIPLES AND ART OF DIALING. A. DIAL is a plane, upon which lines are de- LECT. scribed in such a manner, that the shadow of a x - wire, or of the upper edge of a plate stile, erected perpendicularly on the plane of the dial, Preiimi- may shew the true time of the day. The edge of the plate by which the time of the day is found, is called the stile of the dial, which must be parallel to the earth's axis ; and the line on which the said plate is erected, is called the substile. The angle included between the substile and stile, is called the elevation, or height, of the stile. Those dials whose planes are parallel to the plane of the horizon, are called horizontal dials^, Vol. II. A 2 Of Dialing. LECT. and those dials whose planes are perpendicular x ' ( to the plane of the horizon, are called vertical, or erect, sun-dials. Those erect dials, whose planes directly front the north or south, are called direct north or south dials ; and all other erect dials are called decliners, because their planes are turned away from the north or south. Those dials, whose planes are neither parallel nor perpendicular to the plane of their horizon, are called inclining, or reclining dials, accord- ing as their planes make acute or obtuse angles with the horizon ; and if their planes are also turned aside from facing the south or north, they are called declining-inclining, or declining- reclining dials. The intersection of the plane of the dial, with that of the meridian, passing through the stile, is called the meridian of the dial, or the hour- line of XII. Those meridians, whose planes pass through the stile, and make angles of 15, 30, 45, 6O, 75, and 90 degrees with the meridian of the place (which marks the hour-line ; of XII) are called hour-circles ; and their intersections with the plane of the dial, are called hour-lines. In all declining dials, the substile makes an angle with the hour-line of XII ; and this angle is called the distance of the substile from the meridian. The declining plane's difference of longi- tude, is the angle formed at the intersection of the stile and plane of the dial, by two meridians ; one of which passes through the hour-line of XII, and the other through the substile. Of Dialing. 3 This much being premised concerning dials in LECT. general, we shall now proceed to explain the dif- x ' ferent methods of their comtruction. If the whole earth, a P c p were transparent, PLATE and hollow, like a sphere of glass, and had its xx - equator divided into 24 equal parts by so many ' g ' *' ! 1 7 J s 9. Theuni- meridian semicircles, a, b, c, a, ey/, g, &c. one versal of which is the geographical meridian of any principle given place, as London, which is supposed to be^ ; ^ ic at the point a; and if the hours of XII were depends; marked at the equator, both upon that meri- dian and the opposite one, and all the rest of the hours in order on the rest of the meridians, those meridians would be the hour-circles of London ; then, if the sphere had an opaque axis, as P E p, terminating in the poles P and />, the shadow of the axis would fall upon every particular meridian and hour, when the sun came to the plane of the opposite meridian, and would consequently shew the time at London, and at all other places on the meridian of Lon- don. If this sphere was cut through the middle by tfr a solid plane ABCD, in the rational horizon of ^ London, one half of the axis EP vould be above the plane, and the other half below it j and if straight lines were drawn from the centre of the plane, to those points where the circumfer- ence is cut by the hour-circles of the sphere, those lines would be the hour-lines of a hori- zontal dial for London : for the shadow of the axis would fall upon each particular hour-line of the dial, when it fell upon the like hour- circle of the sphere. If the plane which cuts the sphere be upright, Fig. 3. at AF'CG* touching the given place (London) at jP, and directly facing the mer:dian of Lon- A ii 4 Of Dialing. LECT. don, it will then become the plane of an erect ^ __, direct south dial ; and if right lines be drawn from its centre , to those points of its circum- fai. ' Ca ference where the hour-circles of the sphere cut it, these will be the hour-lines of a vertical or direct south dial for London, to which the hours are to be set as in the figure (contrary to those on a horizontal dial), and the lower half E p of the axis will cast a shadow on the hour of the day in this dial, at the same time that it would fall upon the like hour-circle of the sphere, if the dial plane was not in the way. inclining If the plane (still facing the meridian) be d ai"" ma< ^ e to incline, or recline, by any given num- ber of degrees, the hour-circles of the sphere will still cut the edge of the plane in those points to which the hour-lines must be drawn straight from the centre ; and the axis of the sphere will cast a shadow on these lines at the Declining respective hours. The like will still hold, if the plane be made to decline by any given number of degrees from the meridian, toward the east or west : provided the declination be less than 90 degrees, or the reclination be less than the co-latitude of the .place : and the axis of the sphere will be a gnomon, or stile, for the dial. But it cannot be a gnomon, when the declina- tion is quite 90 degrees, nor when the reclination is equal to the co-latitude ; ' because in these two cases, the axis has no elevation above the plane of the dial. And thus it appears, that the plane of every dial represents the plane of some great circle 1 If the latitude be subtracted from 90 degrees, the re- mainder is called the co-latitude, or complement of the la- titude. Of Dialing. 5 apon the earth ; and the gnomon the earth's LECT, axis, whether it be a small wire, as in the above , x * figures, or the edge of a thin plate, as in the common horizontal dials. The whole earth, as to its bulk, is but a point, if compared to its distance from the sun ; and therefore, if a small sphere of glass be placed upon any part of the earth's surface, so that its axis be parallel to the axis of the earth, and the sphere have such lines upon it, and such planes within it, as above described, it will shew the hours of the day as truly as if it were placed at the earth's centre, and the shell of the earth were as transparent as glass. But because it is impossible to have a hollow Fig. , 3. sphere of glass perfectly true, blown round a solid plane : or if it was, we could not get at the plane within the glass to set it in any given po- sition ; we make use of a wire sphere to explain the principles of dialing, by joining 24 semi- circles together at the poles, and putting a thin flat plate of brass within it. A common globe, of 1 2 inches diameter, has Dialing by generally 24 meridian semicircles drawn upon 11 " 00 /"" tc L 11-1 i i i i mon terresr it. It such a globe be elevated to the latitude tri of any given place, and turned about until any one of these meridians cuts the horizon in the north point, where the hour of XII is supposed to be marked, the rest of the meridians will cut the horizon at the respective distances of all the other hours from XII. Then, if these points of distance be marked on the horizon, and the globe be taken out of the horizon, and a flat board or plate be put into its place, even with the surface of the horizon, and if straight lines t>e drawn from the centre of the board to those A 3 6 Of Dialing. LECT. points of distance on the horizon which were ,__ '' , cut by the 24 meridian semicircles, these lines will be the hour-lines of a horizontal dial for that latitude, the edge of whose gnomon must , be in the very same situation that the axis of the globe was, before it was taken out of the horizon : that is, the gnomon must make an angle with the plane of the dial equal to the latitude of the place for which the dial is made. If the pole of the globe be elevated to the co-latitude of the given place, and any meridian be brought to the north point of the horizon, the rest of the meridians will cut the horizon in the respective distances of all the hours from XII, for a direct south dial, whose gnomon must make an angle with the plane of the dial, equal to the co-latitude of the place ; and the hours must be set the contrary way on this dial, to what they are on the horizontal. But if your globe have more than twenty- four meridian semicircles upon it, you must take the foljowing method for making horizontal and south dials fry it. To con- Elevate the pole to the latitude of your place, struct a to- and turn the globe until any particular meridian "ia *' (suppose the first) comes to the north point of the horizon, and the opposite meridian will cut the horizon in the south. Then, set the hour- index to the uppermost XII on its circle ; which done, turn the globe westward until fifteen de- grees of the equator pass under the brazen me- ridian, and then the hour-index will be at I (for the sun moves fifteen degrees every hour) and the first meridian will cut the horizon in the number of degrees from the north point, that I is uistant from XII. Turn on until other 15 degrees of the equator pass under the brazen Of Dialing. 7 tneridian, and the hour index will then be at II, LECT. and the first meridian will cut the horizon in , x ' the number of degrees that II is distant from XII: and so, by making 15 degrees of the equator pass under the brazen meridian for every hour, the first meridian of the globe will cut the horizon in the distances of all the hours from XII to VI, which is just 90 degrees ; and then you need go no farther, for the distances of XI, X, IX, VIII, VII, and VI, in the fore- noon, are the same from XII, as the distances of I, II, III, IV, V, and VI, in the afternoon ; and these hour-lines continued through the centre, will give the opposite hour- lines on the other half of the dial : but no more of these lines need be drawn than what answer to the sun's continuance above the horizon of your place on the longest day, which may be easily found by the twenty-sixth problem of the fore- going lecture. Thus to make a horizontal dial for the lati- tude of London, which is 51-^- degrees north, elevate the north pole of the globe 51-^ degrees above the north point of the horizon, and then turn the globe, until the first meridian (which is that of London on the English terrestrial globes) cuts the north point of the horizon, and set the hour-index to XII at noon. Then, turning the globe westward until the index points successively to I, II, III, IV, V, and VI, in the afternoon ; or until 15, 30, 45, 60, 75, and 9O degrees of the equator pass un- der the brazen meridian, you will find that the first meridian of the globe cuts the horizon in the following number of degrees from the north toward the east, viz. 1 If, 24 1, 38 T V, 53f, 7 ', and 90 j which are the respective distances of Of Dialing. LECT. the above hours from XII upon the plane of the horizon. To transfer these, and the rest of the hours, to a horizontal plane, draw the parallel right Fi g- * lines a c and b d upon that plane, as far from each other as is equal to the intended thickness of the gnomon or stile of the dial, and the space included between them will be the meridian or twelve o'clock line on the dial. Cross this meridian at right angles with the six o'clock line g A, and setting one foot of your compasses in the intersection , as a centre, describe the quadrant g e with any convenient radius or opening of the compasses : then, setting one foot in the intersection b, as a centre, with the same radius describe the quadrant fh, and di- vide each quadrant into 9O equal parts or de- grees, as in the figure. Because the hour-lines are less distant from each other about noon, than in any other part of the dial, it is best to have the centres of these quadrants at a little distance from the centre of the dial-plane, on the side opposite to XII, in order to enlarge the hour- distances thereabout under the same angles on the plane. Thus, the centre of the plane is at C, but the centres of the quadrants at a and b. . Lay a ruler over the point b (and keeping it there for the centre of all the afternoon hours in the quadrant fh) 9 draw the hour-line of I, through 1 ly degrees in the quadrant ; the hour- line of II, through 24^ degrees ; of III, through 38 degrees; IIII, through 53^', and V, through 7*1 Y T : and because the sun rises about four in the morning, on the longest days at London, continue the hour-lines of IIII and V, in the afternoon, through the centre b to the opposite i Of Dialing. 9 side of the dial. This done, lay the ruler to the LECT. centre a, of the quadrant e g, and through the , x - like divisions or degrees of that quadrant, viz. llf, 24^, 38-^, 53-f, and 71-jV, draw the fore- noon hour-lines of XI, X, IX, VIII, and VII ; and because the sun does not set before eight in the evening on the longest days, continue the hour- lines of VII and VIII in the forenoon, through the centre , to VII and VIII in the afternoon ; and all the hour-lines will be finished on this dial, to which the hours may be set, as in the figure. Lastly, through 51-^- degrees of either quad- rant, and from its centre, draw the right line a g for the hypothenuse or axis of the gnomon a g i ; and from g, let fall the perpendicular g ?', upon the meridian line a i, and there, will be a triangle made, whose sides are a g, g /, and i a. If a plate, similar to this triangle, be made as thick as the distance between the lines a c and b d, and set upright between them, touching at a and I:, its hypothenuse a g will be parallel to the axis of the world, when the dial is truly set ; and will cast a shadow on the hour of the day. JV. B. The trouble of dividing the two quad- rants may be saved, if you have a scale with a line of chords upon it, such as that on the right hand of the plate ; for if you extend the com- passes from to 6O degrees of the line of chords, and with that extent, as a radius, describe the two quadrants upon their respective centres, the above distances may be taken with the com- passes upon the line, and set off upon the qua- drants. To make an erect direct south dial. Elevate Tocou . the pole to the co-latitude of your place, and tructan proceed in all respects as above taught for th 10 Of Dialing. horizontal dial, from VI in the morning to VI in the afternoon, only the hours must be reversed, as in the figure ; and the hypothenuse a g, of the gnomon a g f, must make an angle with the Fig. s. dial-plane, equal to the co-latitude of the place. As the sun can shine no longer on this dial than from six in the morning till six in the evening, there is no occasion for having any more than twelve hours upon it. * To con- To make an erect dial, declining from the south toward the east or west. Elevate the pole to the ial, latitude of your place, and screw the quadrant of altitude to the zenith. Then, if your dial de- clines toward the east (which we shall suppose it to do at present) count on the horizon the degrees of declination, from the east point to- ward the north, and bring the lower end of the quadrant to that degree of declination at which the reckoning ends. This done, bring any par- ticular meridian of your globe (as suppose the first meridian) directly under the graduated edge of the upper part of the brazen meridian, and set the hour-index to XII at noon. Then, keep- ing the quadrant of altitude at the degree of declination in the horizon, turn the globe east- ward on its axis, and observe the degrees cut by the first meridian in the quadrant of altitude (counted from the zenith) as the hour- index comes to XI, X, IX, &c. in the forenoon, or as ] 5, 3O, 45, &c. degrees of the equator pass un- der the brazen meridian at these hours re- spectively ; and the degrees then cut in the 9 A new and very simple geometrical method of con- structing sun-dials may be seen in our author's Mechanical pxercises, p. 94. ED. Of Dialing. 11 quadrant by the first meridian, are the respective LECT. distances of the forenoon hours from XII on the ( plane of the dial Then, for the afternoon hours, turn the quadrant of altitude round the zenith until it comes to the degree in the ho- rizon opposite to that where it was placed be- fore ; namely, as far from the west point of the horizon toward the south, as it was set at first from the east point toward the north ; and turn th globe westward on its axis, until the first meridian comes to the brazen meridian again, and the hour-index to XII : then, continue to turn the glob-: westward, and as the index points to the afternoon hours I, II, III, &c. or as 15, 30, 45, &c. degrees of the equator pass under the brazen meridian, the first meridian will cut the quadrant of altitude in the respective num- ber of degrees from the zenith, that each of these hours is from XII on the dial. And note, that when the first meridian goes off the quad- rant at the horizon, in the forenoon, the hour- index shews the time when the sun will come upon this dial : and when it goes off the quad- rant in the afternoon, the index will point to the time when the sun goes off the dial. Having thus found all the hour-distances from XII, lay them down upon your dial -plate, either by dividing a semicircle into two quad- rants of 9O degrees each (beginning at the hour- line of XII) or by the line of chords, as above directed. In all declining dials, the line on which the stile or gnomon stands (commonly called the sn/'. clining the same number west. 3, A south dial, declining east. And, 4, A south dial, declining west. Only, placing the proper number of hours, and the stile or gnomon respectively, upon each plane. For, (as above mentioned) in the south- west plane, the substile line falls among the af- ternoon hours ; and in the south-east, of the same declination among the forenoon hours, at equal distances from XII. And so, all the morning hours on the west decliner will be like ihe afternoon hours on the east decliner ; the south-east decliner will produce the north-west decliner ; and the south-west decliner, the north- east decliner, by only extending the hour-lines, stile and subtile, quite through the centre : the axis of the stile, (or edge that casts the shadow on the hour of the day) being in all dials what- ever parallel to the axis of the world, and con- sequently pointing toward the north pole of the heaven in north latitudes, and toward the south pole, in south latitudes. See more of this in the following lecture. But because every one who would like to An easy make a dial, may perhaps not be provided with ^n^^ct a globe to assist him, and may probably not un- ing of * derstand the method of doing it by logarithmic ^ calculation ; we shall shew how to perform it by the plain dialing lines, or scale of latitudes and hours ; such as those on the right hand of Fig. 4, in Plate XXI, or at the top of Plate XXII, and which may be had on scales com- monly sold by the mathematical instrument makers. This is the easiest of all mechanical methods, and by much the best, when the lines are truly 14 Of Dialing. LECT. divided : not only the half hours and quarters , ^ , may be laid down by ail of them, but tvery fifth minute by most, and every single minute by those where the line of hours is a foot in length. rig. 3. Having drawn your double meridian line a b, cd, on the plane intended for a horizontal dial, and crossed it at right angles by the six o'clock line f e (as in Fig. l), take the latitude of your place with the compasses, in the scale of lati- tudes, and set that extent from c to e, and from a tof) on the six o'clock line : then, taking the whole six hours between the points of the com- " passes in the scale of hours, with that extent set one foot in the point e, and let the other foot fall where it will upon the meridian line c d, as at d. Do the same from f to b, and draw the right lines e d andfb, each of which will be equal in length to the whole scale of hours. This done, setting one foot of the compasses in the beginning of the scale at XII, and extending the other to each hour on the scale, lay off these ex- tents from d to e for the afternoon hours, and from b to f for those of the forenoon : this will divide the lines d e and b f in the same manner as the hour-scale is divided, at 1,2, 3, 4, 5, and 6, on which the quarters may also be laid down, if required. Then, laying a ruler on the point c, draw the first five hours in the afternoon, from that point, through the dots at the numeral figures 1, 2, 3, 4, 5, on the line de ; and con- tinue the lines of IIII and V through the centre c to the other side of the dial, for the like hours of the morning ; which done, lay the ruler on the point a, and draw the last five hours in the forenoon through the dots 5, 4, 3, 2, 1, on the line fb ; continuing the hour-lines of VII Of Dialing. 15 and VIII through the centre a to the other side of the dial, for the like hours of the evening ; and set the hours to their respective lines as in the figure. Lastly, make the gnomon the same way as taught above for the horizontal dial, and the whole will be finished. To make an erect south dial, take the co- latitude of your place from the scale of latitudes, and then proceed in all respects for the hour- lines, as in the horizontal dial ; only reversing the hours, as in Fig. 2 ; and making the angle of the stile's height equal to the co-latitude. I have drawn out a set of dialing lines upon the top of Plate XXII large enough for making a dial of nine inches diameter, or more inches if required ; and have drawn them tolerably ex- act for common practice, to every quarter of an hour. This scale may be cut oft" from the plate, and pasted upon wood, or upon the inside of one of the boards of this book ; and then it will be somewhat more exact than it is on the plate, for being rightly divided upon the copper- plate, and printed off on wet paper, it shrinks as the paper dries ; but when it is wetted again, it stretches to the same size as when newly printed ; and if pasted on while wet, it will re- main of that size afterwards. But lest the young dialist should have neither globe nor wooden scale, and should tear or otherwise spoil the paper one in pasting, we shall now shew him how he may make a dial with- out any of these helps. Only, if he has not a line of chords, he must divide a quadrant into C X) equal parts or degrees for taking the pro- per angle of the stile's elevation, which is easily done. 16 Of Dialing. LECT. With any opening of the compasses, as Z L, . x ' . describe the two semicircles LFk and -LQh, upon the centres Z and z, where the six o'clock line crosses the double meridian line, and divide Herizontat eacn semicircle into 12 equal parts, beginning at fiat. L ; though, strictly speaking, only the quadrants from L to the six o'clock line need be divided ; then connect the divisions which are equidistant from L, by the parallel lines KM, IN, HO, GP, and FQ. Draw FZ for the hypothenuse of the stile, making the angle FZ E equal to the lati- tude of your place ; and continue the line J^Z to R. Draw the line Rr parallel to the six o'clock line, and set off the distance a k from Z to F, the distance b I from Z to X, c H, from Z to W, dG from Z to T, and eFfrom Z to S. Then draw the lines Ss, Tt, Ww, Xx, and Yy each parallel to R r. Set off the distance y Y from a to 11, and from/ to 1 ; the distance x X from b to 1O, and from g to 2 ; w Whom c to 9, and from h to 3 ; t T from d to 8, and from i to 4 ; sS from e to 7 5 and from n to 5. Then, laying a ruler to the centre Z, draw the fore- noon hour lines through the points 11, 10, 9, 8, 7 ; and laying it to the centre z, draw the af- ternoon lines through the points 1 , 2, 3, 4, 5 ; continuing the forenoon lines of VII and VIII through the centre Z, to the opposite side of the dial, for the like afternoon hours ; and the after- noon lines IIII and V through the centre z, to the opposite side, for the like morning hours. Set the hours to these lines as in the figure, and then erect the stile or gnomon, and the horizon- tal dial will be finished. Stuth&j. To construct a south dial, draw the lines VZ, making an angle with the meridian ZZ equal to the co-latitude of your place, and pro- Oj Dialing. 17 ceed in all respects as in the above horizontal dial for the same latitude, reversing the hours as in Fig. 2, and making the elevation of the gnomon equal to the co-latitude. Perhaps it may not be unacceptable to explain the method of constructing the dialing lines, and some others, which is as follows. With any opening of the compasses, as E A, PLATE according to the intended length of the scale, XXIL describe the circle AD CB, and cross it at right Fig. i. angles by the diameters CEA and DEB. Di-f' a/ '" i * n r I lines, how vide the quadrant A B first into nine equal parts, construct- and then each part into 10 ; so shall the quad- ed - rant be divided into 90 equal parts or degrees. Draw the right line AFB for the chord of this quadrant, and setting one foot of the compasses in the point A, extend the other to the several divisions of the quadrant, and transfer these divisions to the line AFB by the arcs, 10 1O, 20 20, &c. and this will be a line of chords divided into QO unequal parts ; which, if trans- ferred from the line back again to the quadrant, will divide it equally. It is plain by the figure, that the distance from A to 60 in the line of chords, is just equal to A E, the radius of the circle from which that line is made ; for if the arc 60 60 be continued, of which A is the centre, it goes exactly through the centre E of the arc A Jj. And therefore, in laying down any number of degrees on a circle, by the line of chords, you must first open the compasses, so as to take in just 60 degrees upon that line, as from A to 6O : and then, with that extent, as a radius, describe a circle which will be exactly of the same size with that from which the line was divided : Vol. II. B 18 Of Dialing. LECT. which done, set one foot of the compasses in the ^ beginning of the chord line, as at A, and extend the other to the number of degrees you want upon the line, which extent, applied to thg circle, will include the like number of degrees upon it. Divide the quadrant CD into 9O equal parts, and from each point of division draw right lines as i, h, /, &c. to the line CE ; all perpendicular to that line, and parallel to D E 9 which will divide E C into a line of sines ; and although these are seldom put among the dialing lines on a scale, yet they assist in drawing the line of latitudes. For, if a ruler be laid upon the point D, and over each division in the line of sines, it will divide the quadrant CE into 9O unequal parts, as B a, a b, &c. shewn by the right lines lOa, 2O b, 30 c, &c. drawn along the edge of the ruler. If the right line B C be drawn, sub- tending this quadrant, and the nearest distances B a, B b, Cc, &c. be taken in the compasses from J5, and set upon this line in the same man- ner as directed for the line of chords, it will make a line of latitudes B C, equal in length to the line of chords AB> and of an equal number of divisions, but very unequal as to their lengths. Draw the right line D GA, subtending the quadrant DA ; and parallel to it, draw the right line r s, touching the quadrant D A at the numeral figure 3. Divide this quadrant into six equal parts, as 1, 2, 3, &c. and through these points of division draw right lines from the centre E to the line r s, which will divide it at the points where the six hours are to be placed, as in the figure. If every sixth part of the quadrant be subdivided into four equal parts, right lines drawn from the centre through these Of Dialing. 19 points of division, and continued to the line r s 9 LECT. will divide each hour upon it into quarters. In Fig. 2, we have the representation of a portable dial, which may be easily drawn on a, card, and carried in a pocket-book. The lines Fg- a d 9 a by and b c, of the gnomon must be cut quite through the card ; and as the end a b of the gnomon is raised occasionally above the plane of the dial, it turns upon the uncut line c d as on a hinge. The dotted line AE must be slit quite through the card, and the thread must be put through the slit, and have a knot tied behind, to keep it from being easily drawn out. On the other end of this thread is a small plum- met _D, and on the middle of it a small bead for shewing the time of the day. To rectify this dial, set the thread in the slit right against the day of the month, and stretch the thread from the day of the month over the angular point where the curve lines meet at XII ; then shift the bead to that point on the thread, and the dial will be rectified. To find the hour of the day, raise the gnomon (no matter how much or how little) and hold the edge of the dial next the gnomon toward the sun, so as the uppermost edge of the shadow of the gnomon may just cover the shadow line ; and the bead then playing freely on the face of the dial, by the weight of the plummet, will shew the time of the day among the hour- lines, as is forenoon or afternoon. To find the time of sun rising and setting, move the thread among the hour-lines, until it either covers some one of them, or lies parallel betwixt any two ; and then it will cut the time of sun-rising among the forenoon hours, and of sun-setting among the afternoon hours, on that B2 20 Of Dialing. day of the year for which the thread Is set in the scale of months. To find the sun's declination, stretch the thread from the day of the month over the angular point at XII, and it will cut the sun's declination, as it is north or south, for that day, in the arched scale of north and south declina-- tion. To find on what day the sun enters the signs : when the bead, as above rectified, moves along any of the curve lines which have the signs of the zodiac marked upon them,- the sun enters those signs on the days pointed out by the thread in the scale of months. The construction of this dial is very easy, especially if the reader compares it all along with Fig. 3, as he reads the following explana- tion of that figure. tig. 3. Draw the occult line AB parallel to the top of the card, and cross it at right angles with the six o'clock line BCD; then upon C, as a centre, with the radius C A, describe the semicircle A EL, and divide it into 12 equal parts (be- ginning at A}, as AT, rs, &c. and from these points of division, draw the hour-lines r, s, t, u, v, E, w, and ar, all parallel to the six o'clock line E C. If each part of the semicircle be divided into four equal parts, they will give the half- hour lines and quarters, as in Fig. 2. Draw the right line ASDo, making the angle SAB equal to the latitude of your place. Upon the centre A describe the arch RST, and set off upon it the arcs SR and S T, each equal to 23- degrees, for the sun's greatest declination ; and divide them tnto 23^- equal parts, as in Fig. 2. Through the intersection D of the lines ECD and A Do the right line FD G at right angles to Of Dialing. 21 A Do. Lay a ruler to the points A and R 9 and LECT. draw the line ARF through 2Z~ degrees of x - south declination in the arc S R ; and then lay- ing the ruler to the points A and 7 1 , draw the line ATG through 23^ degrees of north decli- nation in the arc S T : so shall the linei ARF and ATG cut the line FDG'm the proper length for the scale of months. Upon the centre D, with the radius D F, describe the semicircle Fo G ; and divide it into six equal parts, Fm, m?t, no, &c. and from these points of division draw the right lines m A, n z, p k, and q /, each parallel to o D. Then setting one foot of the compasses in the point F, extend the other to A and describe the arc A z U for the tropic of V? : with the same extent, setting one foot in G, de- scribe the arc AEO for the tropic of 25 . Next setting one foot in the point A, and extending the other to A^ describe the arc AC I for the beginnings of the signs %Z and : ; and with the same extent, setting one foot in the point /, de- cribe the arc AN for the beginnings of the signs n and Q.. Set one foot in the point z, and having extended the other to A, describe the arc AKfor the beginnings of the signs X and HI ; and with the same extent, set one foot in h, and describe the arc A M for the beginnings of the signs tf and H. Then, setting one foot in the point Z), and extending the other to A^ describe the curve A L for the beginnings of T and tQ: ; and the signs will be finished. This done, lay a ruler from the point A over the sun's declina- tion in the arch RST (found by the following table) for every fifth day of the year :and where the ruler cuts the line FDG 9 make marks ; and place the days of the month right against these marks, in the manner shewn by Fig. 2* B 3 22 Of Dialing. LECT. Lastly, draw the shadow line P Q parallel to the _^ occult line AE ; make the gnomon, and set the hours to their respective lines, as in Fig. 2, and the dial will be finished. Fig. 4- There are several kinds of dials, which are called universal, because they serve for all lati- tudes. Of these, the best one that I know is Mr. Pardie's, which consists of three principal parts : the first whereof is called the horizontal An univen- plane (A] because in the practice it must be aidM. parallel to the horizon. In this plane is fixed an upright pin, which enters into the edge of the second part D, called the meridional plane ; which is made of two pieces, the lowest whereof (5) is called the quadrant, because it contains a quarter of a circle, divided into 9O degrees ; and it is only into this part, near B, that the pin enters. The other piece is a semicircle (D) adjusted to the quadrant, and turning in it by ^ groove, for raising or depressing the diameter (jE F) of the semicircle, which diameter is called the axis of the instrument. The third piece is a circle (G) divided on both sides into 24 equal parts, which are the hours. This circle is put upon the meridional plane so, that the axis (EF) may be perpendicular to the circle ; and the point C be the common centre of the circle, semicircle, and quadrant. The straight edge of the semicircle is chamfered on both sides to a sharp edge, which passes through the centre of the circle. On one side of the chamfered part, the first six months of the year are laid down, according to the sun's declination for their respective days, and on the other side the last six months. And against the days on which the sun enters the signs, there are straight lines drawn upon the semicircle, with the Of Dialing. 23 characters of the signs marked upon them. LECT. There is a black line drawn along the middle of the upright edge of the quadrant, over which hangs a thread (//) with its plummit (/) for levelling the instrument. N. B. From the 22 d of September to the 20 th of March, the upper surface of the circle must touch both the centre C of the semicircle, and the line of V and * ; and from the 2O th of March to the 22 d of Sep- tember, the lower surface of the circle must touch that centre and line. To find the time of the day by this dial. Having set it on a level place in sun-shine, and adjusted it by the levelling screws k and /, until the plumb line hangs over the back line upon the edge of the quadrant, and parallel to the said edge ; move the semicircle in the quadrant until the line of T and * (where the circle touches) comes to the latitude of your place in the quadrant : then, turn the whole meri- dional plane B D, with its circle G, upon the horizontal plane A, until the edge of the shadow of the circle falls precisely on the day of the month in the semicircle ; and then, the me- ridional plane will be due north and south, the axis E F will be parallel to the axis of the world, and will cast a shadow upon thfe true time of the day, among the hours on the circle. JV. B. As, when the instrument is thus recti- fied, the quadrant and semicircle are in the plane of the meridian, so the circle is then in the plane of the equinoctial : therefore, as the sun is above the equinoctial in summer (in northern latitudes) and below it in winter ; the axis of the semicircle will cast a shadow on the hour of the day, on the upper surface of the circle, from the 2O th of March to the 22 d of September ; 24 Of Dialing. LECT. and from the 22 d of September to the 20 th of x - March, the hour of the day will be determined ""*"""-' by the shadow of the semicircle, upon the lower surface of the circle. In the former case, the shadow of the circle falls upon the day of the month, on the lower part of the diameter of the semicircle ; and in the latter case on the upper part. The method of laying down the months and signs upon the semicircle, is as follows. Draw the right line ACB, equal to the diameter of Fig, 5. the semicircle AD B, and cross it in the middle at right angles with the line E CD, equal in length to ADB ; then E C will be the radius of the circle F C G, which is the same as that of the semicircle. Upon E as a centre, describe the circle FCG, on which set off the arcs Ch and Ci 9 each equal to 23-f degrees, and divide them accordingly into that number for the sun's declination. Then, laying the edge of a ruler over the centre E, and also over the sun's de- clination for every fifth day 3 of each month (as in the card-dial), mark the points on the dia- meter AB of the semicircle from a to g, which are cut by the ruler ; and there place the days of the months accordingly, answering the sun's declination. This done, setting one foot of the compasses in C, and extending the other to a or g, describe the semicircle abed efg ; which divide into six equal parts, and through the points of division draw right lines, parallel to CD, for the beginning of the signs (of which one half are on one side of the semicircle, and J The intermediate days may be drawn in by hand, if the spaces be large enough to contain them. Of Dialing. 25 the other half on the other side), and set the characters of the signs to their proper lines, as in the figure. The following table shews the sun's place and Tables rf declination, in degrees and minutes, at the noon thc sun>s of every day of the second year after leap year -, L which is a mean between those of leap year it- self, and the first and third years after it. It is useful for inscribing the months and their days on sun-dials ; and also for finding the latitudes of places, according to the methods prescribed after the table. * 4 In this edition, the tabk of the sun's longitude and declination has been calculated anew, and adapted to the present improved state of the solar tables. The editor has also added an accurate table of the equation of time, which, he trusts, will be of great use to the practical dialist. The signs + and , add aud subtract, at the head of the column, denote that the equation of time must be added to, -or subtracted from, the apparent time, or that which it deduced from the motion of thc sun, in order to obtain the equated or true time, as shewn by a well-regulated clock or watch. The table is calculated for thc second after leap year, and is as accurate as the difference bxtweea thc civil and solar year will permit. Ep. 26 Tables of the Sun's Place and Declination. A Table shewing the sun's place and declination. January. February. B Sun's PI. Sun's Dec. Sun's PL Sun's Dec. co D. M. D. M. CO D. M. D. M. 1 10v? 27 23 S 3 1 1 O/Wf f\ L J./VW U 17S 13 2 11 28 22 58 2 13 1 16 56 3 12 29 22 53 3 J4 2 16 38 4 13 31 22 47 4 15 3 16 2O 5 14 32 22 40 5 16 4 16 2 6 15 33 22 34 6 17 4 15 44 7 16 34 22 26 7 18 5 15 26 8 17 35 22 19 8 19 6 15 7 9 18 37 22 1O 9 2O 7 14 48 10 19 38 22 2 10 21 7 14 28 11 2O 39 21 53 11 22 8 14 9 12 21 40 21 43 12 23 9 13 49 13 22 41 21 S3 13 24 13 29 14 23 42 21 23 14 25 10 13 9 15 24 43 21 12 15 26 10 12 49 16 25 44 21 1 16 27 11 12 28 17 26 46 20 5O 17 28 11 12 7 18 27 47 20 38 18 29 12 11 46 19 28 48 2O 25 19 0X12 11 25 2O 29 49 2O 13 20 1 12 11 3 21 OZZ5O 2O O 21 2 13 1O 42 22 1 51 19 46 22 3 13 10 2O 23 2 52 19 32 23 4 13 9 58 24 3 53 19 18 24 5 14 9 36 25 4 54 19 4 25 6 14 9 14 26 5 55 18 49 26 7 14 8 52 27 6 56 18 34 27 8 14 8 29 28 7 57 18 18 28 9 15 8 7 29 8 58 18 2 In these Tables N sig- 30 9 58 17 46 nines north, and S 31 1O 59 17 3O south, declination. Tables of the Sun's Place and Declination. 27 A Table shewing the sun's place and declination. ' March. April. O ^< O) Sun's PL Sun's Dec C ^ ^ r Sun's PI. Sun's Dec. D. M. D. M. D. M D. M. 1 1OX15 7S44 1 11T 3 4N23 2 11 15 7 21 2 12 2 4 46 3 12 15 6 58 3 13 1 5 9 4 13 15 6 35 4 14 O 5 32 5 14 15 6 -J2 5 14 59 5 55 6 15 15 5 49 6 15 58 6 17 7 16 15 5 26 7 16 57 6 4O 8 17 15 5 2 8 17 56 7 3 9 18 15 4 39 9 18 55 7 25 10 ]9 15 4 16 10 19 54 7 47 11 20 15 3 52 11 2O 53 8 9 12 21 15 3 29 12 21 51 8 31 13 22 14 3 5 13 22 50 8 53 14 23 14 2 41 14 23 49 9 J5 15 24 14 2 18 15 24 47 9 37 16 25 13 4S 54 16 25 46 9 58 17 26 13 3fr 30 17 26 44 10 19 18 27 12 1 7 18 27 43 10 40 19 28 12 O 43 19 28 41 11 1 20 29 11 O 19 20 29 4O 11 22 21 OT11 ON 4 21 Otf 38 11 43 22 1 10 28 22 4i- 37 12 3 23 2 10 52 23 2 35 12 23 24 3 9 I 15 24 3 33 12 43 25 26 4 9 5 8 1 3Q 2 2 25 26 4 32 5 3O 13 3 13 22 27 6 7 2 26 27 6 28 13 42 28 7 6 2 50 28 fo 27 14 1 29 8 6 3 13 29 8 25 14 2O 30 3* 9 5 1O 4 3 36 3 59 SO 9 23 14 38 LECT. 28 Tables of the Sun's Place and Declination. LECT. X. A Table shewing the sun's place and declination. May. June. i Sun's PL Sun's Dec. o Sun's PL bun's Dec. D. M. D. M. CO D. M. D. M. i 10X21 14N57 1 ion 12 22N O 2 11 19 15 15 2 11 10 22 8 3 12 18 15 33 3 12 7 22 16 4 13 16 15 50 4 13 4 22 24 5 14 14 16 8 5 14 2 22 31 6 15 12 16 25 6 14 59 22 37 7 16 10 16 42 7 15 57 22 43 8 17 8 16 58 8 16 54 22 49 9 18 6 17 14 9 17 51 22 55 10 19 4 17 30 10 18 49 23 O 11 20 2 17 46 11 19 46 23 4 12 20 59 18 1 12 2O 43 23 9 13 21 57 18 17 13 21 4O 23 12 14 22 55 18 31 14 22 38 23 16 15 23 53 18 46 15 23 35 23 19 16 24 51 19 o 16 24 32 23 21 17 25 48 19 14 17 25 29 23 23 18 26 46 19 27 18 26 27 23 25 19 27 44 19 41 19 27 24 23 26 20 28 41 19 53 20 28 21 23 27 21 29 39 20 6 21 29 18 23 28 22 o!nS7 20 18 22 0.2516 23 28 23 1 34 20 30 23 1 13 23 28 24 2 32 20 41 24 2 10 23 27 25 3 29 20 53 25 3 7 23 26 26 4 27 21 3 26 4 5 23 24 27 5 24 21 14 27 5 2 23 22 28 6 22 21 24 28 5 59 23 20 29 7 20 21 33 29 6 56 23 17 3O 8 17 21 43 30 7 53 23 14 31 9 15 21 52 Tattes Of the Sun's Place and Declination. 29 A Table shewing the sun's place and declination. July. August. O Sun's PI. Sun's Dec. Sun's PI. Sun's Dec. *|j D. M. D. M. ^ D. M. D. M. 1 85551 23N1O 1 80,26 18N10 2 9 48 23 6 2 9 24 17 55 3 10 45 23 2 3 10 21 17 40 4 11 42 22 57 4 11 19 17 24 5 12 40 22 52 5 12 16 17 8 6 13 37 22 46 6 13 14 16 52 7 14 34 22 40 7 14 11 16 35 8 15 31 22 34 8 15 9 16 19 9 16 28 22 27 9 16 6 16 2 10 17 26 22 20 10 17 4 15 44 11 18 23 22 12 11 18 2 15 27 12 19 2O 22 4 12 18 59 15 9 13 20 17 21 56 13 19 57 14 51 14 21 14 21 47 14 20 54 14 33 15 22 12 21 38 15 21 52 14 14 16 23 9 21 29 16 22 50 13 55 17 24 6 21 19 17 23 47 13 36 18 25 3 21 9 18 24 45 13 17 19 26 1 2O 58 19 25 43 12 58 20 26 58 20 47 2O 26 41 12 58 21 27 55 2O 36 21 27 39 12 18 22 28 52 20 24 22 28 36 11 58 23 29 5O 20 13 23 29 34 11 38 24 00.47 20 O 24 0!t32 11 18 25 1 44 19 48 25 1 SO 10 57 26 2 42 19 35 26 2 28 10 36 27 3 39 19 22 27 3 26 10 15 28 4 37 19 8 28 4 24 9 54 29 5 34 1 8 54 29 5 22 9 33 30 6 31 18 4O 30 6 20 9 12 31 7 29 ,18 25 31 7 18 8 50 Tables of the Sun's Place and Declination. LECT. X. A Table shewing the sun's place and declination. September. October. U Sun's PI. Sun's Dec O Sun's PI. Sun's Dec. GO D. M. D. M. CO D. M. D. M. 1 8tJU7 8N29 1 7:2=34 3S 2 9 15 8 7 2 8 33 3 24 3 10 13 7 45 3 9 33 3 47 4 11 11 7 23 4 1O 32 4 10 5 12 9 7 1 5 11 31 4 34 6 13 8 6 38 6 12 30 4 57 7 14 6 6 16 7 13 29 5 20 8 15 4 5 53 8 14 29 5 43 9 16 2 5 31 9 15 28 6 6 10 17 1 5 8 10 16 27 6 29 11 17 59 4 45 11 17 27 6 51 12 18 58 4 22 12 18 26 7 14 13 19 56 3 59 13 19 26 7 37 14 20 55 3 36 14 2O 25 7 59 15 21 53 3 13 15 21 25 8 22 16; 22 52 2 50 16 22 24 8 44 17 23 50 2 27 17 23 24 9 6 18 24 49 2 4 18 24 23 9 28 19 25 47 1 40 19 25 23 9 50 20 26 46 1 17 20 26 23 1O 11 21 27 45 O 54 21 27 23 1O 33 22 28 44 O 30 22 28 22 1O 54 23 129 42 7 23 29 22 11 16 24 0:0:41 OS 16 24 O"t22 11 37 25 1 4O 40 25 1 22 11 58 26 2 39 1 3 26 2 22 12 18 27 3 38 1 27 27 3 22 12 39 28 4 37 1 50 28 4 22 12 59 29' 5 36 2 14 29 5 22 13 14 30 6 35 2 37 30 6 22 'l3 39 | 31 7 22 13 59 4 Talks of the Sun's Place and Declination. 31 A Table shewing the sun's place and declination. November. December. O CJ Sun's PI. Sun's Dec Sun's PI. Sun's Dec. 3 i ea D. M. D. M. CO D. M. D. M. 1 8m'22 14S 19 1 8^39 21 S46 2 9 22 14 38 2 9 39 21 55 3 1O 23 14 57 3 1O 40 22 4 4 11 23 15 16 4 11 41 22 13 5 12 23 15 34 5 12 42 22 21 6 13 23 15 53 6 13 45 22 28 7 14 24 16 11 7 14 44 22 35 8 15 24 16 28 8 15 45 32 42 9 16 24 16 46 9 16 46 22 48 10 17 24 17 3 1O 17 47 22 54 11 18 25 17 2O 11 18 48 23 12 19 25 17 36 12 19 49 23 5 13 2O 26 17 53 13 20 50 23 9 14 21 26 18 9 14 21 51 23 13 15 22 27 18 24 15 22 52 23 16 16 23 27 18 39 16 23 53 23 19 17 24 28 18 54 17 24 55 23 22 18 25 28 19 9 18 25 56 23 24 19 26 29 19 23 19 26 57 23 26 20 27 SO 19 37 20 27 58 23 27 21 28 30 19 51 21 28 59 23 28 22 29 31 20 4 22 ovy o 23 28 23 O$32 20 17 23 1 2 23 28 24 1 33 2O 30 24 2 3 23 27 25 2 33 20 42 25 3 4 23 26 26 3 34 20 53 26 4 5 23 24 27 4 35 21 5 27 5 6 23 22 28 5 36 21 16 28 6 8 23 19 29 6 37 21 26 29 7 9 .23 16 30 7 38 21 36 30 8 1O 23 13 31 9 11 23 9 LECT. X. 32 Tables of the Equation of Time. Table of the Equation of Time. 4 CO January. February. March. April. M. S. M. S. M. 8. M. S. 1 3 + 48 13 + 58 12 + 45 4+ 7 2 4 16 14 5 f2 33 3 49 3 4 44 14 12 12 21 3 31 4 5 12 14 19 12 8 3 13 5 5 39 14 24 11 54 2 55 6 6 6 14 28 11 40 2 37 7 6 33 14 32 11 26 2 20 8 6 59 14 34 11 14 2 2 9 7 24 14 37 10 56 1 45 10 7 49 14 38 lO 41 1 28 11 8 13 14 39 1O 25 1 12 12 8 37 14 38 10 9 O 55 13 9 14 37 9 52 O 39 14, 9 22 14 35 9 35 23 35 44 14 33 9 18 8 16 1O 4 14 29 9 1 0- 7 17 10 25 14 25 8 43 22 18 1O 44 14 20 8 25 36 19 11 3 14 15 8 7 O 50 20 11 21 14 8 7 49 1 4 21 11 38 14 2 7 31 1 17 22 11 55 13 54 J 12 1 3O 23 12 11 13 46 6 54 1 42 24 12 26 13 37 6 35 1 54 25 12 40 13 28 6 17 2 5 26 12 53 13 18 5 53 2 16 27 13 6 13 8 5 39 2 26 28 13 18 12 57 5 21 2 36 29 13 29 5 2 2 45 30 13 39 ' 4 44 2 53 31 13 49 4 24 Tables of the Equation of Time. 33 Table of the Equation of Time. o 1 y May. June. July. August. M. S. M. S. M. S. M. S. i 3- 2 2 42 3+13 5 + 57 2 3 10 2 34 3 25 5 54 3 3 17 2 24 3 36 5 5O 4 3 23 2 14 3 48 5 46 5 3 29 2 4 3 58 5 4O 6 3 35 1 54 4 9 5 35 7 3 4O 1 43 4 19 5 28 8 3 44 1 32 4 29 5 21 9 3 48 1 21 4 38 5 14 10 3 51 1 10 4 47 5 5 11 3 54 58 4 56 4 57 12 3 56 O 46 5 4 4 47 IS 3 57 O 34 5 11 4 37 14 3 58 22 5 18 4 27 15 3 59 9 5 25 4 16 16 3 59 O + 3 5 31 4 4 17 3 58 O 16 5 37 3 52 18 3 57 O 28 5 42 3 39 19 3 55 O 41 5 47 3 26 20 3 53 O 54 5 51 3 13 21 3 50 1 -7 5 54 2 59 22 3 46 1 2O 5 57 2 45 23 3 42 1 33 6 2 30 24 3 38 1 46 6 2 2 14 25 3 33 1 59 6 3 1 59 26 3 27 2 11 6 4 1 43 27 3 21 2 24 6' 5 1 26 28 3 14 2 37 6 4 1 9 29 3 7 2 49 6 3 52 SO 3 59 3 1 6 2 O 34 31 2 51 6 17 Vol. II. 34 Talks of the Equation of Time. LECT. x. Table of the Equation of Time. d P 3 a Septem. October. Novem. Decem. M. S. M. S. M. S. M. S. 1 0- 2 1010 1613 1049 2 O 2O 10 29 16 14 1O 27 3 39 10 47 16 14 1O 3 4 O 58 11 6 16 14 9 39 5 1 18 11 24 16 13 9 15 6 1 37 11 42 16 11 8 5O 7 1 57 11 59 16 8 8 24 8 2 18 12 16 16 4 7 58 9 2 38 12 32 16 7 31 1O 2 58 12 48 15 54 7 4 11 3 19 13 4 15 48 6 37 12 3 40 13 19 15 41 6 9 13 4 1 13 84 15 33 5 41 14 4 22 13 48 15 25 5 13 15 4 43 14 2 15 15 4 44 16 5 4 14 15 15 5 4 15 17 5 25 14 27 14 53 3 45 18 5 46 14 39 14 41 3 16 19 6 7 14 5.O 14 28 2 46 20 6 28 15 1 14 14 2 16 21 6 4Q 15 11 13 59 1 46 22 7 10 15 20 13 44 1 16 23 7 30 15 28 13 27 45 24 7 51 15 36 13 10 15 25 8 11 15 43 12 52 0+15 26 8 32 15 50 12 33 45 27 8 52 15 55 12 14 1 15 28 9 12 16 11 54 1 45 29 9 31 16 5 11 33 2 14 30 9 51 16 8 11 12 2 43 31 16 11 3 13 35 Explanation of the Table of the Equation of Time. As our author has already given a familiar explana- tion of the equation of time, it may be sufficient to ob- serve, that the preceding table contains the difference between true and apparent time, for every day of the year at 12 o'clock noon, when the sun is in the meridian ; and is adapted to the second year after leap year. If apparent, or solar, time is to be converted into true time, as shewn by a well-regulated clock or watch, the equa- tion of time must be added to the apparent time, if it has the sign -f-, and subtracted from it if it has the sign : but if true is to be converted into apparent time, the equation must be applied with contrary signs. If the equation is required for any intermediate hour, take the difference during a day, and say, as 24 hours is to this difference, so is the number of hours which the in- termediate hour is from the preceding noon, to a third proportional, which, added to, or subtracted from, the equation of time at noon, according as it is increasing or decreasing, will give the equation of time for the given hour. If the equation of time is wanted, at a time when the signs change from + to , or from to +, the difference for 24 hours will be found by adding the equations of time for the noon preceding and following the given hour. Thus, if the equation of time is required for the 24 th December at 12 o'clock midnight, the equation for the 24 th at noon is 15", and for the 25 th at noon 4. 15", the difference ot which is + 30". Then, as 24 h : + 30" = 12 h : + 15, which, subtracted from 15 seconds, because the numbers are decreasing, the equation for the 24 th noon, leaves 0, so that the hour, as shewn by the sun and clock, is the same on the 24 tb December at midnight. The equation thus found will be accurate for every second year after leap year, and in other years will vary only a few se- conds from the truth. In order, however, to determine the equation of time, with accuracy for any other year, Ca 36 find the difference between the equation of time for the given day, and that which precedes it : then, 1. For leap year , take one half of this difference, and add it to the equation for the given time if it in- creases, but subtract it if it decreases. 2. For thejirst after leap year, take one fourth of the difference, and add it to the equation for the given time if it increases, but subtract it if it decreases. 3. For the third after leap year y take one fourth of the difference, and subtract it from the equation for the given time, if it increases, but add it if it decreases. Thus, to find the equation of time for the 2 d May 1805, being the first after leap year, the equation in the table is 3 10", the daily difference is 8", and the equation increases. Add, therefore, 2", which is one fourth of the daily difference, to 3 10", and the sum 3' 12", will be the true equation of time for the 2* Mny 1805. Ru es for finding the Latitude* 37 TO FIND THE LATITUDE OF ANY PLACE BY OBSER- VATION. The latitude of any place is equal to the ele- vation f of the pole above the horizon of that place. Therefore it is plain, that if a star was fixed in the pole, there would be nothing re- quired to find the latitude, but to take the alti- tude of that star with a good instrument. But although there is no star in the pole, yet the la- titude may be found by taking the greatest and least altitude of any star that never sets : for if half the difference between these altitudes be add- ed to the least altitude, or subtracted from the greatest, the sum or remainder will be equal to the altitude of the pole at the place of observation. 1 But because the length of the night must be more than 12 hours, in order to have two such observations ; the sun's meridian altitude and de- clination are generally made use of for finding the latitude, by means of its complement, which is equal to the elevation of the equinoctial above the horizon ; and if this complement be sub- tracted from 9O degrees, the remainder will be the latitude, concerning which, I think, the fol- lowing rules take in all the various cases. 1 . If the sun has north declination, and is on the meridian, and to the south of your place, subtract the declination from the meridian alti- tude (taken by a good quadrant) and the remain- 1 If the altitude of the pole star be taken six horn's be fore, or after, it comes to the meridian, or arrives at iti point of greatest and least altitude, the latitude of ths place will thus be accnratcly obtained by only one observa- tion ED. 38 Rules for finding the Latitude. LECT. der will be the height of the equinoctial or com- * plement of the latitude north. EXAMPLE. w f The sun's meridian altitude 42 20 1 " south. 56 1 And his declination, subtract 10 15 north. Remains the complement of the latitude, 32 5 Which subtract from 90 And the remainder is the latitude 57 55 north. * 2. If the sun has south declination, and is southward of your place at noon, add the decli- nation to the meridian altitude ; the sum, if less than 90 degrees, is the complement of the lati- tude north : but if the sum exceeds 90 degrees, the latitude is south ; and if 90 be taken from that sum, the remainder will be the latitude. EXAMPLES. The sun's meridian altitude 65 10' south The sun's declination, add 15 30 south Complement of the latitude 80 40 Subtract from 90 Remains the latitude 9 20 north The sun's meridian altitude 80 40 7 south The sun's declination, add 20 10 south The sum is 100 50 From which subtract 90 Remains the latitude 10 50 so * The sun's meridian altitude, as taken by a quadrant, or any other instrument, must be corrected by the appli- cation of parallax and refraction. As the sun is elevated by refraction and depressed by parallax, his apparent me- ridian altitude must be diminished by the difference between the refraction and parallax, -En, Rules fot finding the Latitude. 39 3. If the sun has north declination, and is on the meridian north of your place, add the decli- nation to the north meridian altitude ; the sum, if less than 90 degrees, is the complement of the latitude south ; but if the sum is more than 90 degrees, subtract 90 from it, and the remainder is the latitude north. EXAMPLES. Sun's meridian altitude 60 30' north Sun's declination, add 20 10 north Complement of the latitude 80 40 Subtract from 90 Remains the latitude 9 20 south Sun's meridian altitude 70 20' north Sun's declination, add 23 20 north The sum is 93 40 From which subtract 90 Remains the latitude 3 40 north 4. If the sun has south declination, and is north of your place at noon, subtract the decimation from the north meridian altitude, and the re- mainder is the complement of the latitude south* EXAMPLE. Sun's meridian altitude 52* 30* north Sun's declination, subtract 20 10 south Complement of the latitude 32 20 Subtract from 90 And the remainder is the latitude 57 40 4O Rules for finding the Latitude* , 5. If the sun has no declination, and is south of your place at noon, the meridian altitude is the complement of the latitude north : but if the sun be then north of your place, his meridian altitude is the complement of the latitude south. EXAMPLES. Sun's meridian altitude 38 3& south Subtract from 90 Remains the latitude 51 30 north Sun's meridian altitude 38 30' north Subtract from 90 Remains the latitude 51 30 south 6. If you observe the sun beneath the pole, subtract his declination from 90 degrees, and add the remainder to his altitude ; and the sum is the latitude. / EXAMPLE. Sun's declination 20 30' Subtract from 90 Remains 69 30 \ ,, Sun's altitude below the pole 10 20/ c The sum is the latitude 79 50 Which is north or south, according as the sun's declination is north or south : for when the sun has south declination, he is never seen below the north pole ; nor is he ever seen below the south pole, when his declination is north. Rules for Jinding the Latitude. 41 7. If the sun be in the zenith at noon, and at the same time has no declination, you are then under the equinoctial, and so have no latitude. If the sun be in the zenith at noon, and has declination, the declination is equal to the lati- tude, north or south. These two cases are so plain, that they require no examples. 3 3 The latitude of a place may be found with equal fa- cility and accuracy, by taking the meridian altitude of the planets and fixed stars, and observing the same directions which are given by our author in the case of the sun. When fixed stars, however, are employed, their altitude must be corrected by refraction only> as their parallax is not sensible. En. ; * LECTURE XI. OF DIALING. IJ.AVING shewn in the preceding Lecture how to make sun-dials by the assistance of a good globe, or of a dialing scale, we shall now pro- ceed to the method of constructing dials arith- metically ; which will be more agreeable to those who have learned the elements of trigonometry, because globes and scales can never be so accu- rate as logarithms, in finding the angular dis- tances of the hours. Yet, as a globe may be found exact enough for some other requisites in dialing, we shall take it in occasionally. The construction of sun-dials on all planes whatever, may be included in one general rule : intelligible, if that of a horizontal dial for any given latitude be well understood. For there is no plane, however obliquely situated with re- spect to any given place, but what is parallel to the horizon of some other place j and therefore, if we can find that other place by a problem on the terrestrial globe, or by a trigonometrical cal- culation, and construct a horizontal dial for it ; that dial, applied to the plane where it is to Of Dialing. 43 serve, will be a true dial for that place. Thus, LECT. an erect direct south dial in 51^- degrees north xl - latitude, would be a horizontal dial on the same meridian, 90 degrees southward of 51^- degrees north latitude ; which falls in with 3&f degrees of south latitude : but if the upright plane de- clines from facing the south at the given place, it would still be a horizontal plane 90 degrees from that place, but for a different longitude ; which would alter the reckoning of the hours ac- cordingly. CASE I. J. Let us suppose that an upright plane at London declines 36 degrees westward from fac- ing the south ; and that it is required to find a place on the globe, to whose horizon the said plane is parallel ; and also the difference of lon- gitude between London and that place. Rectify the globe to the latitude of London, and bring London to the zenith under the brass meridian, then that point of the globe which lies in the horizon at the given degree of declination (counted westward from the south point of the horizon) is the place at which the above-mention- ed plane would be horizontal. Now, to find the latitude and longitude of that place, keep your eye upon the place, and turn the globe eastward until it comes under the graduated edge of the brass meridian ; then the degree of the brass meridian that stands directly over the place, is its latitude ; and the number of degrees in the equator, which are intercepted between the me- ridian of London and the brass meridian, is the place's difference of longitude. Thus, as the latitude of London is 51^ de- 44 Of Dialing. LECT. grees north, and the decimation of the place is 36 XL degrees west ; I elevate the north pole 51 de- " J grees above the horizon, and turn the globe un- til London comes to the zenith, or under the graduated edge of the meridian ; then, I count 36 degrees on the horizon westward from the south point, and make a mark on that place of the globe over which the reckoning ends, and bringing the mark under the graduated edge of the brass meridian, I find it to be under 30^ de- grees in south latitude : keeping it there, I count in the equator the number of degrees between the meridian of London and the brasen meridian (which now becomes the meridian of the requir- ed place) and find it to be 42 1. Therefore an upright plane at London, declining 36 degrees westward from the south, would be a horizontal plane at that place ; whose latitude is 30^ degrees south of the equator, and longitude 42^ degrees west of the meridian of London. Which difference of longitude being convert- ed into time, is 2 hours 5\ minutes. The vertical dial declining westward 36 de- grees at London, is therefore to be drawn in all respects as a horizontal dial for south latitude 30^ degrees ; save only, that the reckoning of the hours is to anticipate the reckoning on the horizontal dial, by 2 hours 51 minutes : for so much sooner will the sun come to the meridian of London, than to the meridian of any place whose longitude is 42 1 degrees west from London. * LATE 2. But to be more exact than the globe will xxiii. shew us, we shall use a little trigonometry. Let NE SIP be the horizon of London, whose zenith is Z, and P the north pole of the sphere ; and let Z h be the position of a vertical plane at Of Dialing. 45 Z, declining westward from S (the south) by LF.CT an angle of 36 degrees ; on which plane an erect dial for London at Z is to be described. Make "^ the semidiameter Z D perpendicular to Z A, and it will cut the horizon in ), 36 degrees west of the south S. Then, a plane in the tangent ///), touching the sphere in _D, will be parallel to the plane Z h ; and the axis of the sphere will be equally inclined to both these planes. Let ff^QE be the equinoctial, whose elevation above the horizon of Z (London) is 38|- de- grees ; and PRD be the meridian of the place D, cutting the equinoctial in R. Then, it is evi- dent, that the arc RD is the latitude of the place D (where the plane Z h would be horizontal) and the arc RQ is the difference of longitude of the planes Z h and DH. In the spherical triangle WDR^ the arc IVD is given, for it is the complement of the plane's declination from S the south ; which comple- ment is 54 (viz. 90 36) : the angle at /?, in which the meridian of the place D cuts the equa- tor, is a right angle ; and the angle RWD mea- sures the elevation of the equinoctial above the horizon of Z, namely 38^- degrees. Say, there- fore, as radius is to the co-sine of the plane's de- clination from the south, so is the co-sine of the latitude of Z to the sign of RD the latitude of D ; ' which is of a different denomination from the latitude of Z, because Z and D are on dif- ferent sides of the equator. As radius , ~ * 10.00000 Toco-sine 36 V=RQ 9.90796 So co-sine 51 3Q'=QZ 9.79415 To sine 30 U'=DR 9.70211=* 1 See Playfair's Hements of Geometry. Spher, Trig*. Prop. XIX. ED. 46 Of Dialing. LECT. the latitude of D, whose horizon is parallel to XL the vertical plane Z h at Z. V^MMM^ N. B. When radius is made the first term, it may be omitted, and then, by subtracting it, mentally, from the sum of the other two, the operation will be shortened. Thus, in the pre- sent case, To the logarithmic sine of WR= * 54 - 0' 9.90796 Add the logarithmic sine of RD= S 38 3(X 9.79415 Their sum minus radius 9.7021 1 gives the same solution as above. And we shall keep to this method in the following part of the work. To find the difference of longitude of the places D and Z, say, as radius is to the co-sine of 38-i- degrees, the height of the equinoctial at Z, so is the co-tangent of 36 degrees, the plane's declination, to the co-tangent of the difference of longitudes. 6 Thus, To the logarithmic sine of 7 51 3& 9.89364 Add the logarithmic tang, of 8 54 w 0' 10.13874 Their sum minus radius _ . . . , 10.03238 is the nearest tangent of 47 8'ir WR ; which is the co-tangent of 42 52' RQ, the difference of longitude songht. Which difference being re- duced to time, is 2 hours 51^ minutes. 3. And thus having found the exact latitude 4 The co-sine of 36 O', or of * The co-sine of 51 3O / , or of jZ. ' Playfair's Geom. Spher. Trig. Prop. XVIII ED- 7 The co-sine of 38 30', or of WDR. 8 The co-tangeat of 36, or of DW. Of Dialing. 47 and longitude of the place Z), to whose horizon LECT. the vertical plane at Z is parallel, we shall pro- Xi - ceed to the construction of a horizontal dial for the place Z), whose latitude is 30 1 4' south ; but anticipating the time at D by 2 hours 51 minutes (neglecting the -^ minute in practice) be- cause D is so far westward in longitude from the meridian of London; and this will be a true vertical dial at London, declining westward 36 degrees. Assume any right line C S L for the substile Fig. a. of the dial, and make the angle KC P equal to the latitude of the place (viz. 30 1 4 ) to whose horizon the plane of the dial is parallel ; then C R P will be the axis of the stile, or edge that casts the shadow on the hours of the day, in the dial. This done, draw the contingent line E Q, cutting the substilar line at right angles in K; and from K make KR perpendicular to the axis C R P. Then KG (=KR) being made radius, that is equal to the chord of 60 or tangent of 45, on a good sector take 42 52' (the differ- ence of longitude of the places Z and Z)) from the tangents, and having set it from K to M y draw CM for the hour-line of XII. Take KN equal to the tangent of an angle less by 1 5 de- grees than KM; that is, the tangent 27 52'; and through the point N draw CN for the hour- line of I. The tangent of 12 J 52' (which is 1 5 less than 27* 52') set off the same way, will give a point between K and JV, through which the hour-line of II is to be drawn. The tangent of 2 8' (the difference between 45 and 42 52') placed on the other side of CZ, will determine ihe point through which the hour-line of III is to be drawn : to which 2 8' if the tangent of 15 be added, it will make 17 8'; and this set off from A!" toward Q on the line .EQ.will give 48 Of Dialing. LICT. the point for the hour-line of IV ; and so of the __, rest. The forenoon hour-lines are drawn the same way, by the continual addition of the tangents 15, 30, 45, &c. to 42% 52 4 (= the tangent of KM) for the hours of XI, X, IX, &c. as far as necessary ; that is, until there be five hours on each side of the substile. The sixth hour, ac- counted from that hour or part of the hour on which the substile falls, will be always in a line perpendicular to the substile, and drawn through the centre C. 4. In all erect dials, CM, the hour-line of XII is perpendicular to the horizon of the place for which the dial is to serve : for that line is the in- tersection of a vertical plane with the plane of the meridian of the place, both which are per- pendicular to the plane of the horizon : and any line HO, or ho, perpendicular to CM, will be a horizontal line on the plane of the dial, along which line the hours may be numbered : and CM being set perpendicular to the horizon, the dial will have its true positron. 5. If the plane of the dial had declined by an equal angle toward the east, its description would have differed only in this, that the hour-line of XII would have fallen on the other side of the substile C L, and the line H would have a sub- contrary position to what it has in this figure. 6. And these two dials, with the upper points of their stiles turned toward the north pole, will serve for the other two planes parallel to them ; the one declining from the north toward the east, and the other from the north toward the west, by the same quantity of angle. The like Of Dialing. 49 holds true of all dials in general, whatever be LECT. their declination and obliquity of their planes to L _ the horizon. CASE n. 7. If the plane of the dial not only declines, Fig. 3. but also reclines, or inclines. Suppose its de- clination from fronting the south S be equal to the arc S D on the horizon $ and its reclination be equal to the arc D d of the vertical circle DZ ; then it is plain, that if the quadrant of altitude ZdD, on the globe, cuts the point D in the horizon, and the reclination is counted upon the quadrant from D to d ; the intersec- tion of the hour-circle P Rd, with the equinoc- tial /^Q, will determine Rd, the latitude of the place d, whose horizon is parallel to the given plane Z h at Z ; and R Q will be the dif- ference in longitude of the planes at d and Z. Trigonometrically thus : let a great circle pass through the three points IV, d, E ; and in the triangle IV D d, right-angled at D, the sides IV D and D d are given ; and thence the angle D IV d is found, and so is the hypothenuse IV d. Again, the difference, or the sum, of D W d, and D IV R, the elevation of the equinoctial above the horizon of Z, gives the angle d JV R ; and the hypothenuse of the triangle iVRdwas just now found ; whence the sides R d and IV R are found, the former being the latitude of the place d, and the latter the complement of R Q, the difference of longitude sought. Thus, if the latitude of the place Z be 52 1 Cf north ; the declination S D of the plane Z h (which would be horizontal at d) be 36, and the reclination be 1 5, or equal to the arc D d ; rhe south latitude of the place r/, that is, the arc R d* 50 Of Dialing. LECT. will be 1,5 9' ; and R Q the difference of the ^__ ^ ( longitude, 36 2'. From these data, therefore, let the dial (Fig. 4) be described, as in the form- er example. 8. Only it is to be observed, that in the reclining or inclining dials, the horizontal line will not stand at right angles to the hour-line of XII, as in erect dials ; but its position may be found as follows. Fig. 4. To the common substilar line C K Z-, on which the dial for the place d was described, draw the dial Cr p m 1 2 for the place Z), whose declina- tion is the same as that of d, viz. the arc S D ; and H 0, perpendicular to C m, the hour-line of XII on this dial, will be a horizontal line on the dial CPRM XII. For the declination of both dials being the same, the horizontal line remains parallel to itself, while the erect position of one dial is reclined or inclined with respect to the position of the other. Or, the position of the dial may be found by applying it to its plane, so as to mark the true hour of the day by the sun, as shewn by another dial ; or, by a clock regulated by a true meridian line and equation table. 9. There are several other things requisite in the practice of dialing ; the chief of which I shall give in the form of arithmetical rules, simple and easy to those who have learned the elements of trigonometry. For in practical arts of this kind, arithmetic should be used as far as it can go ; and scales never trusted to, except in the final construction, where they are abso- lutely necessary in laying down the calculated hour-distances on the plane of the dial. And although the inimitable artists of this metropolis have no occasion for such instructions, yet they Of Dialing. 51 may be of some use to students, and to private LECT. gentlemen, who amuse themselves this way. v- _ _. RULE I. To find the angles which the hour-lines on any dial make with the substile. To the logarithmic sine of the given latitude, or of the stile's elevation above the plane of the dial, add the logarithmic tangent of the hour distance ' from the meridian, or from the sub- stile ; * and the sum minus radius will be the lo- garithmic tangent of the angle sought. tor, in Fig. 2, K C is to K M in the ratio compounded of the ratio of K C to KG (rrAT/2) and of KG to KM; which making CAT the ra- dius, 1O,OOO,OOO, or 100,000, or 1O or 1, are the ratio of 1O,OOO,OOO, or of 1OO,OOO, or of 10, or of 1, to KG x KM. Thus, in a horizontal dial, for latitude 51 30*, to find the angular distance of XI in the forenoon, or 1 in the afternoon, from XII. To the logarithmic sine of 51 3C' 9.8935-1* Add the logarithmic tang, of 51 G' 9.42805 The sum minus radius is , 9.32! 59= the logarithmic tangent of 11 50*, or of the angle which the hour-line of XI or I makes with the hour of XII. 1 That is, of 15, 30, 45, 60, 75 > for the hours of I, IT, III, IV, V, in the afternoon ; and XI, X, IX, VIII, VII, in the forenoon. 1 In all horizontal dials, and erect north or south dials, the substile and meridian are the same ; but in all declin- ing dials, the substile line makes an angle with the meridian. J In which case, the radius C K is supposed to be di- vided into 1,000,000 equal parts. D 2 52 Of Dialing. And by computing in this manner, with the sine of the latitude, and the tangents of 30, 45, 60, and 75, for the hours of II, III, IV, and V, in the afternoon ; or of X, IX, VIII, and VII, in the forenoon ; you will find their angular distances from XII to be 24 18', 38 3', 53 35', and 71 6': which are all that there is oc- casion to compute for. And these distances may be set off from XII by a line of chords ; or ra- ther, by taking 1 ,OOO from a scale of equal parts, and setting that extent as a radius from C to XII : and then, taking 2O9 of the same parts (which, in the tables, are the natural tangent of 11 50'), and setting them from XII to XI and to I, on the line h o, which is perpendicular to rig. a. C XII : and so for the rest of the hour-lines, which in the table of natural tangents, against the above distances, are 451, 782, 1,355, and 2,920, of such equal parts from XII, as the ra- dius C XII contains 1 ,OOO. And lastly, set of 1,257 (the natural tangent of 51 P 30'} for the angle of the stile's height, which is equal to the latitude of the place. The reason why I prefer the use of the tabular numbers, and of a scale decimally divided, to that of the line of chords is because there is the least chance of mistake and error in this way ; and likewise, because in some cases it gives us the advantage of a nonius division. 4 4 This scale, for sub-diyiding the limbs of quadrants, and the divisions of other mathematkal instruments, is im- properly called Nonius, from one Nonius, who is supposed to be its inventor. The honour of the invention is due to Peter Vernier, a French gentleman, from whom it fre- quently receives its name. The Vernfer scale consists of a piece of brass or ivory, which moves along the limb of the quadrant. A space, equal to any number of degrees in the circular arch, 1 1, for example., is transferred to this piece of Of Dialing. .73 in the universal ring-dial, for instance, the LECT. divisions on the axis are the tangents of the^ angles of the sun's declination placed on either side of the centre. But instead of laying them down from a line of tangents, I would make a scale of equal parts, whereof 1 ,OOO should answer exactly to the length of the semi-axis, from the centre to the inside of th equinoctial ring ; and then lay down 434 of these parts toward each end from the centre, which would limit all the divi- sions on the axis, because 434 is the natural tangent of 23 '29'. And thus by a jionius affix- ed to the sliding piece, and taking the sun's de- clination from an ephemeris, and the tangent of that declination from the table of natural tangents, the slider might be always set true to within two minutes of a degree. And this scale of 434 equal parts might be placed right against the 23^- degrees of the sun's declination, on the axis, instead of the sun's of brass, and divided into 10 parts, so that each division of the vernier will exceed each division of the limb by ^ ot a degree. Suppose the plumb-line of the quadrant to fall between the 25 tu and 2(j th degree, and that the degrees run from right to left. Then, in order to find the num- ber of minutes above 25, move the vernier till the plumb- line falls on the beginning of its scale, and find what divi- sion of the vernier coincides with any division on, the limb; and by so many lO' h " of a degree will the angle exceed 25. If the / th division of the vernier, for instance, coin- cides with a division on the limb, then, ,V hf f a De- gree, or 42 minutes, must be added to 25 degrees, and the angle will be 25 42'.- Nonius's method consisted in drawing a number of concentric circles, the outermost of which was divided into yo parts ; the next into 8p ; the next into 88, &c so that the plumb-line was sure to coin- cide with some division in one of these circles, and the angle could be easily deduced, from the number of parts into which that circle was divided. ED. D 3 Of Dialin. 6" place, which is there of very little use. For then, the slider might be set in the usual way, to the day of the month, for common use ; but to the natural tangent of the declination, when great accuracy is required. The like may be done wherever a scale of sines or tangents is required on any instrument. RULE II. The latitude of the place, the sun's declination^ and his hour distance from the meridian being ^ to find ( 1 ,) his altitude; (2,) his azimuth. 1 Let d be the sun's place, d R, his declina- tion : and in the triangle P Z d, P d the sum, or the difference, of d #, and the quadrant P It being given by the supposition, as also the com- plement of the latitude P Z, and the angle dPZ, which measures the horary distance of d from the meridian ; we shall (by case 4, of Keill's Oblique spheric trigonometry) find the base Z d, which is the sun's distance from the zenith, or the com- plement of his altitude. And (2,) as sine Z d : sine P d : : sine d P Z : d Z P, or of its supplement D Z S, the azimuthal distance from the south. Or, the practical rule may be as follows : Write A for the sine of the sun's altitude, L and I for the sine and co-sine of the latitude, D and d for the sine and co-sine of the sun's declination, and Hfor the sine of the horary distance from VI, Then the relation of H to A will have three varieties. 1 . When the declination is toward the elevated pole, and the hour of the day is between XII and VI ; it is A - L D -f- H I d, and H^ Of Dialing. ' 55 V. When the hour is after VI, it is A L D LECT. D jf xi. Hid, and H= l -j- ' 3. When the declination is toward the de- pressed pole, we have A = H I d L D, and E7 A+LD. Id Which theorems will be found useful, and ex- peditious enough for solving those problems in geography and dialing, which depend on the re- lation of the sun's altitude to the hour of the day. EXAMPLE I. Suppose the latitude of the place to be 51-^ degrees north ; the time five hours distant from XII, that is, an hour after VI in the morning, or before VI in the evening ; and. the sun's declina- tion 20 north. Required the sun's altitude ? Then, to log. L = log. sine 51 38' 1.893.54 s Add log. D = log. sine 20 U 0' 1.53405 Their sum 1 .42759 gives L D logarithm of O.267664, in the na- tural sines. And, to log. //= log. sine 5 15 0' 1.41300 ,, / log. /. = log. sine 7 38 0' 1.79414 \ log. d. = log. sine 8 70 0' 1.97300 Their sum 1.18014 gives H I d logarithm of O.15140 y in the nar turai sines. 5 Here we consider the radius as unity, and not 1,000,000, by which, instead of the index 0,, we have 1, as above : which is of no further use, than making the work a little easier. The distance of one hour from VI. ' The co-latitude of the place. * The co-declination of the sun. 56 Of Dialing. And these two numbers of (0.267664 and. 0.151408) make 0.419072 = A ; which, in the table, is the nearest natural sine of 24 47', the sun's altitude sought. The same hour-distance being assumed on the other side of VI, then L D /// d is O.I 16256, the sine of 6 40'-f ; which is the sun's altitude at V in the morning, or VII in the evening, when his north declination is 2O. But when the declination is 20 south, (or to- ward the depressed pole) the difference H I d L D becomes negative, and thereby shews that, an hour before VI in the morning, or past VI in the evening, the sun's centre is 6^ 40' below the horizon. EXAMPLE II. In the same latitude and north declination from the 'given altitude to find the hour. Let the altitude be 48 ; and because, in this case H ~ ld '- and A (the natural sine of 48) = .743145, and LD rr .267664-, AL D will be O.475481, whose logarithmic sine is 1.6771331 from which taking the logarithmic sine of / 4- d = 1.7671354 Remains 1 .9099977 the logarithmic sine of the hour-distance sought, viz. of 54 22' ; which, reduced to time, is 3 h 37^-*; that is, IX * 37-^ in the forenoon, or II h oo-L ra in the afternoon. Put the altitude 18, whose natural sine is .3090170; and thence A L D xvill be = .0491953 ; which divided by / -f d, gives Of Dialing. 57 ,0717179, the sine of 4 6'^ in time 16^- mi- LECT. nutes nearly, before VI in the morning, or af- ( * L ter VI in the evening, when the sun's altitude is 18. And, if the declination 20 had been toward the south pole, the sun would have been de- pressed 18 below the horizon at 16-^- minutes after VI in the evening ; at which time the twi- light would end ; which happens about the 22 d of November, and 19 th of January, in the lati- tude of 51-f north. The same way may the end of twilight, or beginning of dawn, be round "for any time of the year. NOTE 1, If in theorem 2 and 3 (page 55) ^is put O, and the value of H is computed, we have the hour of sun-rising and setting for any latitude, and time of the year. And if we put H O, and compute A^ we have the sun's altitude or depression at the hour of VI. And lastly, if f/ 9 A^ and D are given, the latitude may be found by the resolution of a quadratic equa- tion ; for I = v/ 1 /,*. . NOTE 2, When A is equal 0, H is equal 77 T L x T"Z>, the tangent of the latitude multiplied by the tangent of the declination. As, if it was required, ivhat is the greatest length of day in latitude 51 30' ? To the log. tangent of 51 30' 0.0993948 Add the log. tangent of 23 39' 1 .6379563 rr.i i. ' ' f ? . fi ) 1 1 heir sum 1.7373511 is the log. sine of the hour-distance 33 7' ; in time 2 h m m . The longest day therefore is f 58 Ofbialing. 12 h -f 4 s 25 m 16 h 25 m . And the shortest day is 12 h 4 h 25 m =: 7 " 35 m . And if the longest day is given, the latitude TJ of the place is found ; .^73 being equal to T 1 L. Thus, if the longest day is 13-|- h zz 2 X 6 h -f 45 ra and 45 m in time being equal to 1 1^ degrees. From the log. sine of 1 1 15' 1 .2902357 Take the log. tang, of 23 2{/;' !/J n. : NOTE 3. The same rule for finding the long- est day, in a given latitude, distinguishes the hour-lines that are necessary to be drawn on any dial from those which would be superfluous. In lat. 52 1O' the longest day is 16 h 32 ra and the hour-lines are to be marked from 44 m after III in the morning, to 16 m after VIII in the evening. In the same latitude, let the dial of Art. 7, Fig. 4, be proposed ; and the elevation of its stile (or the latitude of the place d, whose hori- zon is parallel to the plane of the dial) being 15 9' ; the longest day at d, that is, the longest time that the sun can illuminate the plane of the dial, will (by the rule H = T L X T D) be twice 6 h 27 m 12 h 54 m . The difference of longitude of the planes d and Z was found in the same example to be 36 2' j in time. Of Dialing. 59 2 * 24 m ; and the declination of the plane was LECT. from the south toward the west. Adding there- XI - fore 2 to 24 m to 5 h 33 m , the earliest sun-rising 1 ^ on a horizontal dial at d, the sum 7 h 57 m shews that the morning hours, or the parallel dial at Z, ought to begin at 3 m before VIII. And to the latest sun-setting at d t which is 6 h 27 m , adding the same 2 b 24 m , the sum 8 h 51 m exceeding 6 h l6 ra , the latest sun-setting at Z, by 35 m , shews that none of the afternoon hour- lines are superfluous. And the 4 h 13 m from. III h 44 m , the sun-rising at Z, to VII h 57 m ; the sun-rising at d, belong to the other face of the dial ; that is, to a dial declining 36 from north to east, and inclining 15. EXAMPLE III. from the same data to Jind the sun's azimutJi. If //, Z,, and Z), are given, then (by Art. 2, of Rule II) from //, having found the altitude and its complement Z d ; and the arc P D (the distance from the pole) being given, say, as the co-sine of the altitude is to the sine of the distance from the pole, so is the sine of the hour-distance from the meridian to the sine of the azimuth distance from the meridian. ^ Let the latitude be 51 SO' north, the decima- tion 15 9* south, and the time IP 24 m in the afternoon, when the sun begins to illuminate a vertical wall, and it is required to find the po- sition of the wall. Then, by the foregoing theorems, the com- plement of the altitude will be 81 g 32 y, and Of Dialing. IECT. P d the distance from the pole being 109 5', X J] , and the horary distance from the meridian, or the angle d P Z, 36. To log- sine 74 51' ]. 98464 Add log. sine 36 0' 1.76922 And from (he sum 1 .7.5386 Take the log. sine 81 32'| 1.99323 Remains 1.73861 = log. sine 35, the azimuth distance south. When the altitude is given, find from thence the hour, and proceed as above. This praxis is of singular use on many oc- casions : in finding the declination of vertical planes more exactly than in the common way, especially if the transit of the sun's centre is ob- served by applying a ruler with sights, either plane or telescopical, to the wall or plane, whose declination is required. In drawing a meridian- line, and finding the magnetic variation. In finding the bearings of places in terrestrial sur- veys ; the transit of the sun over any place, or his horizontal distance from it being observed, together with the altitude and hour. And thence determining small differences of longi- tude. In observing the variation at sea, &c. The learned Mr. Andrew Reid invented an instrument several years ago, for finding the la- titude at sea from two altitudes of the sun, ob- served on the same day, and the interval of the observations, measured by a common watch. And this instrument, whose only fault was that of its being somewhat expensive, was made by Mr. Jackson. Tables have been lately computed for that purpose. But we may often, from the foregoing rules, resolve tfre same problem without much trouble ; Of Dialing. 6l especially if we suppose the master of the ship LECT. to know within 2 or 3 degrees what his latitude ,__ is, Thus, Assume the two nearest probable limits of the latitude, and by the theorem J7 ~\ /* , com- pute the hours of observation for both supposi- tions. If one interval of those computed hours coincides with the interval observed, the ques- tion is solved. If not, the two distances of the intervals computed, from the true interval, will give a proportional part to be added to, or sub- tracted from, one of the latitudes assumed. And if more exactness is required, the operation may be repeated with the latitude already found. But whichever way the question is solved, a proper allowance is to be made for the difference of latitude arising from the ship's course in the time between the two observations. Of the double horizontal Dial, and the Babylo- nian and Italian Dials. To the gnomonic projection, there is sometimes added a stereographic projection of the hour- circles, and the parallels of the sun's declination, on the same horizontal plane ; the upright side of the gnomon being sloped into an edge, standing perpendicularly over the centre of the projec- tion : so that the dial being in its due position t the shadow of that perpendicular edge is a ver- tical circle passing through the sun, in the ste- reographic projection. The months being duly marked on the dial, the sun's declination, and the length of the day at any time, are had by inspection : as also hk 62 Of Dialing. LECTV altitude, by means of a scale of tangents. But ,_ _^ , its chief property is, that it may be placed true, whenever the sun shines, without the help of any other instrument. E e- 3- Let d be the sun's place in the stereographic projection, x dy z the parallel of the sun's decli- nation, r Ld a vertical circle through the sun's centre, P d the hour-circle ; and it is evident, that the diameter N-S of this projection being placed duly north and south, these three circles will pass through the point d. And therefore, to give the dial its due position, we have only to turn its gnomon toward the sun,, on a horizontal plane, until the hour on the common gnomonic projection coincides with that marked by the hour-circle P d, which passes through the inter- section of the shadow Z d with the circle of the sun's present declination. The Babylonian and Italian dials reckon the hours, not from the meridian, as with us, but PLATE from the sun's rising and setting. Thus, in Italy, one hour before sun-set is reckoned the 23 d hour, two hours before sun-set the 22 d hour, and so of the rest. And the shadow that marks them on the hour-lines, is that of the point of a stile. This occasions a perpetual variation be- tween their dials and clocks, which they must correct from time to time, before it arises to any sensible quantity, by setting their clocks so much faster or slower. And in Italy they begin their day, and regulate their clocks, not from sun-set, but from about mid- twilight, when the Ace Maria is said ; which corrects the difference that would otherwise be between the clock and the dial. The improvements which have been made in all sorts of instruments and machines for mea- Of Dialing. 63 suring time, have rendered such dials of little LECT. account. Yet, as the theory of them is inge- . XI ' nious, and they are really, in some respects, the best contrived of any for vulgar use, a general idea of their description may not be unaccept- able. Let Fig. 5 represent an erect direct south Fig. 5. wall, on which a Babylonian dial is to be drawn, shewing the hours from sun-rising ; the lati- tude of the place, whose horizon is parallel to the wall, being equal to the angle KCR. Make, as for a common dial, KG KR, (which is per*- pendicular to CR^) the radius of the equinoctial JE Q, and draw RS perpendicular to CK for the stile of the dial ; the shadow of whose point R is to mark the hours, when SR is set upright on the plane of the dial. Then it is evident, that in the contingent line M Q, the spaces Kl, K2, K3, &c. being taken equal to the tangents of the hour-distances from the meridian, to the radius KG, one, two, three, &c. hours after sun -rising, on the equinoctial day ; the shadow of the point R will be found, at these times, respectively in the points 1, 2, 3, &c. Draw, for the like hours after sun-rising, when the sun is in the tropic of Capricorn v? v, the like common lines CD, CE, CF, &c. and at these hours the shadow of the point R will be found in those lines respectively. Find the sun's altitudes above the plane of the dial at these hours, and \vith their co-tangents Sd, Sc, Sf^ &c. to radius S R, describe arcs intersecting the hour-lines in the points d, e f, &c. so shall the right lines 1 d, 2 e, 3f, &c. be the lines of I, II, 111, &c. hours after sun-rising. The construction is the same in every other * ' C4 Of Dialing. LECT. case, due regard being had to the difference ojf XI< longitude of the place at which the dial would be horizontal, and the place for which it is to serve. And likewise, taking care to draw no lines but what are necessary ; which may be done, partly by the rules already given for de- termining the time that the sun shines on any plane, and partly from this, that on the tropic- al days the hyperbola described by the shadow of the point R limits the extent of all the hour- lines. The most useful, however, as well as the simplest, of such dials, is that which is described dfi the two sides of the meridian plane. That the Babylonian and Italic hours are truly enough marked by right lines, is easily shewn. Mark the three points on a globe, where the horizon cuts the equinoctial, and the two tropics, toward the east or west : and turn the globe on its axis 15, or J hour; and it is plain that the three points which were in a great circle (viz. the horizon) will be in a great circle still ; which will be projected geometri- cally into a straight line. But these three points are universally the sun's places, one hour after sun-set (or one hour before sun- rise) on the equinoctial and solstitial days. The like is true of all other circles of decimation, beside the tropics ; and therefore, the hours on such dials are truly marked by straight lines limited by the projections of the tropics; and which are rightly drawn, as in the foregoing example. Note ]. The same dials may be delineated without the hour-lines, CD, CE, CF, &c. by setting off the sun's azimuths on the plane of the dial, from the centre S, on either side of the substile GSK, and the corresponding co* Of Dialing. 65 tangents of altitude from the same centre S t for I, II, III, &c. hours before or after the sun is in the horizon of the place for which the dial is to serve, on the equinoctial and solstitial days. 2. One of these dials has its name from the hours being reckoned from sun-rising, the be- ginning of the Babylonian day. But we are not thence to imagine that the equal hours, which it shews, were those in which the astronomers of that country marked their observations. These, we know with certainty, were unequal, like the Jewish, as being twelfth parts of the natural day : and an hour of the night was, in like manner, a twelfth part of the night ; longer or shorter, according to the season of the year. So that an hour of the day, and an hour of the night, at the same place, would always make ~ of 24, or 2 equinoctial hours. In Palestine, among the Romans, and in several other coun- tries, 3 of these unequal nocturnal hours were a vigi'ia, or watch. And the reduction of equal and unequal hours into one another is extreme- ly easy. If, for instance, it is found, by a fore- going rule, that in a certain latitude, at a given time of the year, the length of a day is 14 equi- noctial hours, the unequal hours is then -f^- or f an hour, that is, 70 minutes ; and the nocturnal hour is 50 minutes. The first watch begins at VII (sun-set) ; the second at three times 50 mi- nutes after, viz. IX b 30 m ; the third always at midnight j the morning watch at half an hour past II. If it were required to draw a dial for shewing these unequal hours, or twelfth parts of the day, we must take as many declinations of the sun as are thought necessary, from the equator toward each tropic : and having computed the sun's if. E Of Dialing. altitude and azimuth for -j^, --, ^~ b parts, &c, of each of the diurnal arcs belonging to the de- clinations assumed : by these, the several points in the circles of declination, where the shadow of the stile's point falls, are determined ; and curve lines drawn through the points of a ho- mologous division will be the hour-lines re- quired. * Of the right placing of dials, and having a true meridian line for the regulation of clocks and watches. * The plane on which the dial is to rest, being duly prepared, and every thing necessary for 1 For the description of a new dial, invented by Lam- bert, and of a curious Analemmatic dial, which can be properly placed without a mariner's needle, or a meridian line, and which can be drawn in a garden, the spectator being its stile, see Appendix ED. 1 In another work, when speaking upon the placing of sun-dials, our author observes, ' that if the dial be made according to the strictest rules of calculation, and be truly set at the instant when the sun's centre is on the meridian, it will be a minute too fast in the forenoon, and a minute too slow in the afternoon, by the shadow of the stile ; for the edge of the shadow that shews the time is even with the sun's foremost edge all the time be- fore noon, and even with his hindermost all the afternoon on the dial. And it is the sun's centre that determines the time in the supposed hour-circles of the heavens. And as the sun is half a degree in breadth, he takes two minutes to move through a space equal to his breadth, so that there will be two minutes at noon in which the shadow will have no motion at all on the dial ; conse- quently, if the dial be set true by the sun in the fore- noon, it will be two minutes too slow in the afternoon ; and How to draw a Meridian Line. 67 fixing it, you may find the hour tolerably exact LECT. by a large equinoctial ring-dial, and set your watch to it. And then the dial may be fixed by the watch at your leisure. If you would be more exact, take the sun's altitude by a good quadrant, noting the precise time of observation by a clock or watch. Then, compute the time for the altitude observed (by the rule, page 57), and set the watch to agree with that time, according to the sun. A Had- ley's quadrant is very convenient for this purpose ; for, by it you may take the angle between the sun and his image, reflected from a bason of water : the half of which angle, subtracting the refraction, is the altitude required. This is best done in summer, and the nearer the sun is to the prime vertical (the east or west azimuth) when the observation is made, so much the better. Or, in summer, take two equal altitudes of the sun in the same day ; one any time between seven and ten in the morning, the other be- tween two and five in the afternoon ; noting the moments of these two observations by a clock and if it be set true in the afternoon, it will be two mi- nutes too fast in the forenoon. The only way that I * know of to remedy this is, to set every hour and miaute * division on the dial one minute nearer 12 than the cal- 1 culation makes it to be. Tables and Tracts, 2 a edit. p. 73. These observations are new, and just enough in them- selves ; but the evil which the author points out may be remedied by observing the middle of the shadow's penum- bra, which corresponds with the sun's centre, instead of the border of the real shadow ; and I believe it will be found, that every person naturally does this when he de- termine* the hour of the day upon a sun-dial. ED. F. 2 68 How to draiv a Meridian Line. LECT. or watch : and if the watch shews the observa- ^ r tions to be at equal distances from noon, it agrees exactly with the sun ; if not, the watch must be corrected by half the difference of the forenoon and afternoon intervals ; and then the dial may be set true by the watch. Thus, for example, suppose you have taken the sun's altitude when it was twenty* minutes past VIII in the morning by the watch, and found, by observing in the afternoon, that the sun had the same altitude ten minutes before IV, then it is plain, that the watch was five minutes too fast for the sun : for five minutes after XII is the middle time between VIII h 20 m in the morning, and III h 50 m in the afternoon ; and therefore, to make the watch agree with the sun, it must be set back five minutes. 3 A meridian A good meridian line, for regulating clocks or watches, may be had by the following method. Make a round hole, almost a quarter of an inch diameter, in a thin plate of metal ; and fix the plate in the top of a south window, in such a manner, that it may recline from the zenith at an angle equal to the co-latitude of your place, as nearly as you can guess ; for then the 5 The above method of finding the hour of the day by corresponding altitudes of the sun or stars, is the easiest and most correct that can be employed. Owing, how- ever, to the change that takes place in the sun's declina- tion before the afternoon altitude is taken, it is liable to an error, which, at a maximum, amounts to 301 in the time of the equinoxes. A table containing this cor- rection, which depends upon the interval between the alti- tudes, and upon the declination of the sun, may be seen in the Astronomic de la Lande, edit. 3 , torn, i, Tables, p. 37, and in the Tables de Berlin, torn, i, p. 201. How to draw a Meridian Line. 09 plate will face the sun directly at noon on the LF.CT equinoctial days. Let the sun shine freely through the hole into the room ; and hang a plumb-line to the ceiling of the room, at least five or six feet from the window, in such a place as that the sun's rays, transmitted through the hole, may fall upon the line when it is noon by the clock ; and having marked the said place on the ceiling, take away the line. Having adjusted a sliding bar to a dove-tail groove, in a piece of wood about eighteen inches long, and fixed a hook in the middle of the bar, nail the wood to the above-mentioned place on the ceiling, parallel to the side of the room in which the window is ; the groove and bar be- ing toward the window. Then hang the plumb- line upon the hook of the bar, the weight or plummit reaching almost to the floor ; and the whole will be prepared for farther and proper adjustment. This done, find the true solar time by either of the two last methods, and thereby regulate your clock. Then, at the moment of next noon by the clock, when the sun shines, move the sliding bar in the groove until the shadow of the plumb-line bisects the image of the sun (made by his rays transmitted through the hole) on the floor, wall, or on a white screen placed on the north side of the line ; the plummet or weight at the end of the line hanging freely in a pail of water placed below it on the floor. But because this may not be quite correct for the first time, on account that the plummet will not settle immediately, even in water ; it may be farther corrected on the following days, by the above method, with the sun and clock, and so brought to a very great exactness. E 3 70 How to draw a Meridian Line. LECT. j\r g t T^ ra y s transmitted through the hole ' . will cast but a faint image of the sun, even on, a white screen, unless the room be so darkened that no sun-shine may be allowed to enter but what comes through the small hole in the plate. And always, for some time before the observa- tion is made, the plummet ought to be immers- ed in a jar of water, where it may hang freely ; by which means the line will soon become steady, which otherwise would be apt to con- tinue swinging. As this meridian line will not only be suf- ficient for regulating clocks and watches to the true time by equation tables, but also for most astronomical purposes, I shall say nothing of the magnificent and expensive meridian lines at Bologna and Rome, nor of the better mehods by which astronomers observe precisely the tran- sits of the heavenly bodies over the meridian. 4 4 For farther information upon dialing, the reader may consult Orontii Finei opera, Fol. lib. iii. De horologiis Sciothericis a Joanne Voello, Turoni ]609. Horologio- graphia per Sebastianum, Munsterum 1533. Christ. Clavii Bambergensis horologiorum nova descriptio. Demonstra- tio et constructio horologiorum novorum, auctore Georgio Schombergero. Gnomonice Schoner qto. Wolfii oper. Mathemat. torn, ii, p. 787, Ferguson's Select Exercises, Leybourn's Dialing, Leadbetter's Dialing, and an excel- lent treatise by the celebrated Deparcieux, published at the end of his Traite de Trigonometric rectiligne et spherique, This subject is treated more profoundly by M. Sejour, in his Rccheiches sur la Gnomoniqrue, Ij6l> and in his Traitt Anatyique, torn, i, p, 705. LECTURE XII. SHEWING HOW TO CALCULATE THE MEAN TIME OF ANY NEW OR FULL MOON, OR ECLIPSE, FROM THE CREATION OF THE WORLD TO THE YEAR OF CHRIST 5800- IN the following tables, the mean lunation is LECT. V Tf about a 20 tb part of a second of time longer , , than its measure, as now printed in the last edi- tion of my Astronomy ; which makes the differ- O f new and ence of a hour and thirty minutes in 30OO years. ful1 m ns - But this is not material, when only the mean times are required. PRECEPTS. To find the mean time of any New or Full Moon in any given year and month after the Chris- tian &ra. 1. If the given year be found in the third column of the table oj the moon's mean motion from the sun, under the title, years before and after Christ ; write out that year, with the mean motions belonging to it, and thereto join the given month with its mean motions. But, if the given year be not in the table, take out the next lesser one to it that you find, in the same column j *72 Calculation of mean New and Full Moons. LECT. and thereto add as many complete years, as will XIL make up the given year : then, join the month * ' and all the respective mean motions. 2. Collect these mean motions into one sum of signs, degrees, minutes, and seconds ; re- membering that 60 seconds (") make a minute, 60 minutes (') a degree ; 30 degrees ( y ) a sign, and 12 signs ( s ) a circle. When the signs ex- ceed 12, or 24, or 36 (which are whole circles), reject them, and set down only the remainder ; which, together with the odd degrees, minutes, and seconds, already set down, must be reckon- ed the whole sum of the collection. 3. Subtract the result, or sum of this collec- tion, from 12 signs; and write down the re- mainder. Then look in the table under days., for the next less mean motions to this remainder, and subtract them from it, writing down their remainder. This done, look in the table under hours (mark- ed H) for the next less mean motions to this last remainder, and subtract them from it, writing down their remainder. Then look in the table under minutes (mark- ed M) for the next less mean motions to this remainder, and subtract them from it, writing down their remainder. Lastly, look in the table under seconds (mark- ed S) for the next less mean motion to this re- mainder, either greater or less ; and against it you have the seconds answering thereto. 4. And these times collected, will give the mean time of the required new moon; which will be right in common years j and also in January Calculation of mean New andfull Moons. 73 and February in leap years ; but always one day LECT. too late in leap years after February. . _ xli EXAMPLE i. Required the time of new moon in September 1 764. ? (A year not inserted in the table.) Moon from sun. ISfftllt'frt f-ff; To the year after Christ's birth 1753 10 9 24 56 Add complete years 11 10 14 20 (sum 1764) And join September 2 22 21 8 The sum of these mean motions is 112 24 Which, being subtracted from a circle, or 12 000 leaves remaining 10 27 59 36 Next less mean motion for twenty-six days, subtract 10 16 57 34 And there remains 1 2 2 Next less mean motion for two hours, subtract. 1 57 And the remainder will be 1 5 less mean motion for two minutes, subtract . . 11 Remains the mean motion of twelve se- conds, 4 These times, being collected, would shew the mean time of the required new moon in Septem- ber 1764, to be on the 2y th day, at 2 h 2 m 12 s past noon. But, as it is in a leap year, and after February, the time is one day too late. So, the true mean time is September the 25 th , at 2 m 12* past II in the afternoon. 7 4 Calculation of mean New and Full Moons. N. B. The tables always begin the day at noon, and reckon thenceforward, to the noon of the day following. To find the mean time of full moon in any given year and month after the Christian cera. Having collected the moon's mean motion from the sun for the beginning of the given year and month, and subtracted their sum from twelve signs (as in the former example), add six signs to the remainder, and then proceed in all respects as above. EXAMPLE n. Required the mean time of full moon in September 1764 ? Moon from sun. SO'" To the year after Christ's birth 1753 10 9 24 56 Add complete years 11 10 14 20 (sum 17u'4) And join September 2 22 21 8 The sum of these mean motions is 1 12 24 Which, being subtracted from a circle, or 12 Leaves remaining 10 17 59 36 To which remainder add 6 And the sum will be 4 17 59 36 Next less mean motion for eleven days, subtract , 4 14 5 54 And there remains ................ 3 53 42 Next less mean motion for seven hours, ; ...... ' il subtract ----- ..v-l ...... 3 33 20 And the remainder will be .......... 20 22 Next less mean motion for forty minutes, subtract ...... jl* .vK^ii^^aJi * . . - 20 19 i Remains the mean motion for eight seconds 3 Calculation of mean Neiv and Full Moons. 75 So, the mean time, according to the tables, is the 11 th of September, at 7 " 4Q m 8 s past noon. One day too late, being after February in a leap year. And thus may the mean time of any new or full moon be found, in any year after the Chrisr tian sera. To find the mean time of new or full moon in any given year and month before tlie Christian ccra. If the given year before the year of Christ 1 be found in the third column of the table, under the title of years before and after Christ, write it out, together with the given month, and join the mean motions. But, if the given year be not in the table, take out the next greater one to it that you find ; which being still farther back than the given year, add as many complete years to it as will bring the time forward to the given year ; then join the month, and proceed in all respects as above. EXAMPLE III. Required the mean time of new moon in May^ tfne year before Christ 585 ? The next greater year in the table is 600 ; which being 15 years before the given year, add the mean motions for 1 5 years to those of 6OO, together with those for the beginning of May. 76 Calculation of mean New and Full Moons. Moon from sue, to - ' " LSCT. To the year before Christ 600 511 616 XIL Add complete years motion 15 6 05524 '- ' ' And the mean motion for May 22 53 23 The whole sum is 4 55 3 Which, being subtracted from a circle, or 12 Leaves remaining 11 25 4 57 Next less mean motion for twenty -nine days, subtract 11 23 31 54 And there remains ._. 1 33 3 Next less mean motion for three hours, subtract. 1 31 26 And the remainder will be Next less mean motion for three minutes, subtract . 1 31 Remains the mean motion of fourteen seconds, Q So the mean time, by the tables, was the 29 th of May at 3 h 3 14 s past noon : a day later than the truth, on account of its being in a leap year. For as the year of Christ 1 was the first after a leap year, the year 585 before the year 1 was a leap year of course. If the given year be after the Christian sera, divide its date by 4, and if nothing remains, it is a leap year in the old stile. But if the given year was before the Christian sera (or year of Christ l), subtract one from its date, and divide the remainder by 4 ; then, if nothing remains, it was a leap year ; otherwise not. Calculation of Eclipses. 77 To find whether the sun is eclipsed at the time of any given change, or the moon at any given full. From the table of the sun's mean motion (or LECT. distance) from the moon's ascending node, collect the mean motions answering to the given time ; and if the result shews the sun to be within 1 8 of either of the nodes at the time of new moon, the sun will be eclipsed at that time. Or, if the result shews the time to be within 12 of either of the nodes at the time of full moon, the moon will be eclipsed at that time, in or near the con- trary node j otherwise not. EXAMPLE IV. Q\ , f TV. , The moon changed on the 26'* of September 1764, at 2* 2 OT (neglecting the seconds) afternoon, (See Example I). Qu. JVhether the sun was eclipsed at that time / Sun from node. SO ' " To the year after Christ's birth 1753 1 28 19 Add complete years 11 7 2 3 56 (sum 1764) {September 8 12 22 49 26days 27 13 2houn 5 12 2 minutes 5 Sun's distance from the ascending node 6 9 32 34 Now, as the descending node is just opposite to the ascending (viz. six signs distant from it), and the tables shew only how far the sun has gone from the ascending node, which, by this example, appears to be 6 s 9* 32' 34", it is 78 Calculation of Eclipses. LECT. plain that he must have then been eclipsed j as XIL j he was then only Q 32' 34" short of the des- cending node. EXAMPLE V. The moon was full on the 1 1** of September 1764, at 7 h 40 m past noon. (See Example II.) Qu. Whether she was eclipsed at that time ? Sun from node. so ' " To the year after Christ's birth 1753 1 "28 19 Add complete years 11 7 2 3 56 (sum 1764) fSeptember 8 122249 I 11 days, 11 25 29 } 7hours 18 11 i_ 40minutes 144 Sun's distance from the ascending node 5 24 12 28 Which being subtracted from six signs, leaves only 5 47' 32" remaining ; and this being all the space that the sun was short of the descend- ing node, it is plain that the moon must then have been eclipsed, because she was just as near the contrary node. Calculation of Eclipses. 79 EXAMPLE VI. Q. Whether the sun was eclipsed in May, the year before Christ 585 ? (See Example III.) Sun from node. so it' To the year before Christ 600 9 9 23 51 Add the mean motion of 15 complete years 9 19 27 49 rMay 4 43757 . 29days 1 7 10 And < 3 hours 7 48 3 minutes (neglecting the se- (,_ conds) 8 Sun's distance from the ascending node 3 44 43 Which being less than 1 8, shews that the sun was eclipsed at that time. This eclipse was foretold by Thales, and is thought to be the eclipse which put an end to ecli P sc - the war between the Medes and Lydians. The times of the sun's conjunction with nodes, and consequently the eclipse months of!' any given year, are easily found by the Tables of the sun's mean motion from the moon's ascend- ing node; and much in the same way as the mean conjunctions of the sun and moon are found by the table of the moon's mean motions from the sun. For, collect the sun's mean mo- tion from the node (which is the same as his distance gone from it) for the beginning of any given year, and subtract it from 12 signs ; then, from the remainder, subtract the next less mean motions belonging to whatever month you find them in the table ; and from the remainder sub- tract the next less mean motion for days, and so on for hours and minutes ; the result of all which will shew the time of the sun's mean conjunction with the ascending node of the moon's orbit. 80 To find when there must be Eclipses. EXAMPLE VII. Required the time of the sun's conjunction with the ascending node in the year \ 764 ? ' ? Q (VM f?l 1 ! f" !'*[ lK*i 'i ; Sun from node. so ' " i.iiCT To the year after Christ's birth 1753 1 28 19 xii. Add complete years 11 7 2 3 56 - Mean distance at beginning of A.I) ...1764.. 90 415 Subtract this distance from a circle, or 12000 And there remains 2 29 55 45 Next less mean motion for March, sub- tract 2 1 1639 And the remainder will be 28 39 6 Next less mean motion for 27 days, sub- tract 28 232 ^:V> vA^.vu* -. And there remains 36 34 Next less mean motion for 14 hours sub- tract . 56 21 Remains, nearly, the mean motion of 5 minutes 13 Hence it appears, that the sun will pass by the moon's ascending node on the 27 th of March, at 14 h 5 m past noon, viz. on the 28 tu day, at 5 m past II in the morning, according to the tables; but this being in a leap year, and after February, the time is one day too late. Consequently, the true time is at 5 m past II in the morning on the 27 th day ; at which time the descending node will be directly opposite to the sun. If 6 signs be added to the remainder arising And the remainder will be, 2 29 55 45 To which add half a circle, or 6 And the sum will be 8 29 55 45 JVext less mean motion for September sub- tracted . . 8 12 22 49 And there remains 17 32 56 Next less mean motion for 16 days sub- tracted . 16 37 4 f The Period and Return of Eclipses. 81 from the first subtraction, (viz. from 12 signs) and then the work carried on as in the last xn example, the result will give the mean time of ~" v " the sun's conjunction with the descending node* Thus, in EXAMPLE VIII. To find when the sun will be in conjunction with the descending node in the year 1764 ? Sun from node. so f // To the year after Christ's birth 1753 1 28 19 Add complete years 11 7 2 3 56 Mean distance from ascending node at beginning of 1764 9 4 15 Subtract this distance from a circle, or, 12 And the remainder will be 55 52 iV'xt less mean motion for 21 hours sub- . tracted . 54 32 Remains, nearly, the mean motion of 31 minutes 1 20 So that, according to the tables, the sun will be in conjunction with the descending node on the 16 th of September, at 21 hours 31 minutes FbL If. F 82 T/ie Period and Return of Eclipses. LECT. past noon : one day later than the truth, on ao . XIIt , count of the leap-year. The limits When the moon changes within 18 days be- ofecitfses. fore or after the sun's conjunction with either of the nodes, the sun will be eclipsed at that change : and when the moon is full within 1 2 days before or after the time of the sun's conjunction with either of the nodes, she will be eclipsed at the full : otherwise not. Their pe- if to the mean time of any eclipse, either of the riodandrc- 11 _ _>_ T i , j sun or moon, we add 557 Julian years 21 days 18 hours 11 minutes and 51 seconds (in which there are exactly 689O mean lunations) we shall have the mean time of another eclipse. 5 For at * Dr. HALLEY'S period of eclipses contains only 18 years 1 1 days 7 hours 43 minutes 2O seconds ; in which time, according to his tables, there are just 223 mean lu- nations.: but, as in that time, the sun's mean motion from the node is no more than 11* 29 31' 49", which wants 28' 11'' of being as nearly in conjunction with the same node at the end of the period as it was at the beginning, this period cannot be of constant duration for finding eclips- es, because it will in time fall quite without their limits. The following tables make this period 31" shorter, as ap- pears by the calculation annexed.* The period. Moon from the sun. Sun from node. o / n o / n Complete years 18 7 1 1 5p 41 1 17 46 18 days 114 14 554 112529 hours 7 33320 1811 minutes 42 21 20 1 4$ seconds 44 22 2 Mean motions O 011293149 a By computing from the new solar tables of De Lambrc, and the lunar tables of Mayer, as improved by Mason, this short period of eclipses, which is generally called the period of Pliny, or the Chaldaic period, will amount only to 18 years n days 7 hours 4Z minutes and 31 seconds; and the sun's distance from the moon's node to 8' 10'. EB. The Period and Return of Eclipses. 83 the end of that time the moon will be either new or full, according as we add it to the time of new or full moon ; and the sun will be only 45" farther from the same node, at the end of the said time, than he was at the beginning of it ; as appears by the following example. 6 The period. Moon fronj sup. Sun from node. f 5003 5 32 4710 14 45 8 Complete years < 408 26 50 37 1 23 58 49 (. 173 2 21 3910 28 40 55 days 218 16 21 21 48 38 hours 18 9 8 35 46 44 minutes 11 5 35 29 seconds.. . 51 26 2 Mean motions 00 45 And this period is so very near, that in 60OO years it will vary no more from the truth as to the restitution of eclipses, than 8^ minutes of a degree ; which may be reckoned next to nothing. It is the shortest in which, after many trials, I can find so near a conjunction of the sun, moon, and the same node. 6 The period here mentioned by Mr. Ferguson amounts only to 557 years 21 days 18 hours 4 minutes 47 seconds ; and the sun's distance from the moon's node is fully l' 41''. _ [*; _ I 1 2 84 A Table of Mean Lunations. LECT. This table is made by the continual addition xu - of a mean lunation, viz. 2Q d 12 h 44 ra 3 s 6 th 12 iv "^ ' 14 V 24 vi 0. Lun. Days. H. M. S. Th. In lOOOOO mean luna- tions there are 8085 Ju- 1 29 12 44 3 6 lian years 12 days 21 hours 2 59 1 28 6 13 36 minutes 30 seconds 3 O 88 14 12 9 19 2953059 days 3 hours 36 4 D 118 2 56 12 25 minutes 30 seconds. 5 |- 147 15 40 15 32 Proof of the Table- 6 177 4 24 18 38 Moon fr. sun. 7 2O6 17 8 21 44 In O \ ff 8 236 5 52 24 51 r4ooo 1 14 22 12 9 265 18 36 27 57 cTJ 4000 1 14 22 12 10 295 7 20 31 3 2 ) so 5 23 41 15 20 590 14 41 2 7 3 I 5 10 O 18 28 30 885 22 J 33 11 Days 12 4 26 17 20 40 1181 5 22 4 14 Hours 21 10 40 1 50 1476 12 42 35 18 Min. 36 18 7 100 2953 1 25 10 35 Sec. 20 J5 200 5906 2 50 21 Jl 300 885Q 4 15 31 46 M.fr.sun. O 400 5OO 1OOO 11812 5 40 42 22 14756 7 5 52 57 29530 14 11 45 54 Having by the former precepts computed the mean time of new moon 2OOO 3000 4000 59661 4 23 31 48 88591 18 35 17 42 118122 8 47 3 36 in January, for any given year, it is easy, by this Ta- ble, to find the mean time 5000 10OOO 20000 30000 40OOO 50000 lOOOOO 147652 22 58 49 30 295305 21 57 39 O 590611 19 55 18 885917 17 52 57 O 1181223 15 50 36 1476529 13 48 15 2953059 3 36 39 of new moon in January for any number of years after- wards : and by means of a small table of lunations for 12 or 13 months, to make a general table for finding i-hp mean time of new or full moon in any given year and month whatever. D. H. M. S. Th. In 11 lunations there are* 324 20 4 34 10. In 12 lunations 354 8 48 37 16. la 13 lunations 383 21 32 40 23. But then it would be best to begin the year with March, to avoid the inconvenience of losing a day by mistake in leap year. 85 A Table of the Moon's mean Motion from tiie Sun. Year* Years Years be- Moon from sun Com- Moon from sun. of the of the fore and af- plete Julian World ter Christ. o / n years. , ' period. 706 4008 5 28 ! 1 17 11 1O 14 20 714 8 4000 5 9 23 24 IB 5 2 3 11 1714 1008 3OOO 11 20 28 57 13 9 11 4O 35 2714 2008 2000 6 1 34 30 14 1 21 18 3714 3008 ts iooo O 12 40 3 15 6 O 55 24 3814 3108 '5 900 1O 19 46 36 16 10 22 44 15 3914 3208 O 8OO 8 26 53 9 17 3 2 21 39 4014 3308 *o 700 7 3 59 43 18 7 11 59 4 4114 3408 13 600 5 11 6 16 19 11 21 36 27 4214 3508 500 3 18 12 49 2O 4 13 25 19 4314 3608 400 1 25 19 23 40 8 26 50 37 4414 3708 S 300 2 25 56 60 1 1O 15 56 4514 3808 e 200 1O 9 32 29 80 5 23 41 15 4614 3908 < 100 8 16 39 3 100 10 7 6 33 4714 4008 PQ 1 6 23 45 36 2OO 8 14 13 7 4814 4108 101 5 O 52 9 3OO 6 21 19 40 4914 4208 ; 201 3 7 58 43 400 4 28 26 13 5014 4308 ^ 301 1 15 5 16 500 3 5 32 47 5114 4408 O 401 11 22 11 49 1000 611 5 33 5214 5508 j=f 501 9 29 18 23 200O 22 11 6 5714 5008 & 1001 1 4 51 9 3000 7 3 16 39 6414 5708 1701 O 24 37 2 4OOO 1 14 22 12 6466 576O 1753 10 9 24 56 Vloon from fun. 6514 5808 1801 6 5 26 15 Months. l 1 ' <+ t_ -3 J-H Complete Moon from sun. Jan. O O O C3 r* Ctf b $ years. to ': " Feb. O 17 54 48 cc ^^ """^ 5 > j ^2 -C J3 JZ & i U J3 - 1 4 9 37 24 Mar. 11 29 15 16 U >. -M -G O -r c i^ 2 4 19 14 8 April O 17 1O 3 - C - o3 a> ^j 1 to ou ? C <*- W 28 52 13 May O 22 53 23 IS - 5 . o> o < 4 5 20 41 4 June 1 10 40 11 3*1 . 5 10 18 26 July 1 16 31 32 b 9 8 _ fi m i ^ O C > w ^ 2-0 6 2 9 55 52 Aug. 2 4 6 20 MMi ? O 7 P * X 6 19 33 17 Sept. 2 22 21 b -ill Tgw 8 11 11 22 7 Oct. 2 28 4 29 sSlll JJ 5 9 3 20 59 32 Nov. 3 15 59 I/ o o -= "n TT j= * Th. in i in v , 3 1 6 34 20 ! 30 29 31 15 44 47 4 1 18 43 47 2 1 O 57 32 16 15 16 5 2 O 57 13 3 1 31 26 33 16 45 44 6 2 13 8 4O 4 2 1 54 34 17 16 J3 7 - 2 25 2O 7 5 2 32 23 35 17 46 42 8 3 7 31 34 6 3 2 52 36 18 17 10 9 3 19 43 O 7 3 33 20 37 18 47 39 10 4 1 54 27 8 4 3 49 38 19 18 7 11 4 14 5 54 9 4 34 18 39 19 48 36 12 4 26 17 20 IO 5 4 46 40 20 19 5 13 5 8 28 47 11 5 35 15 41 20 49 33 14 ?5 20 40 14 12 6 5 43 42 21 2O 2 15 6 2 51 40 13 6 36 12 43 21 5O 31 16 6 15 3 7 14 7 6 41 44 22 20 59 17 Q 27 14 34 15 7 37 9 4-5 22 51 28 18 7 26 16 8 7 38 46 23 21 66 19 7 21 37 27 17 8 38 6 47 23 52 25 20 8 3 48 54 18 9 8 36 48 24 22 54 21 8 16 21 19 9 39 4, 49 24 53 22 22 8 28 11 47 20 10 9 32 5O 25 23 51 23 9 IO 23 14 21 IO 4O 1 51 25 54 19 24 9 22 34 41 22 11 IO 30 52 26 24 48 25 10 4 46 7 23 11 40 58 53 26 55 17 26 10 16 57 34 24 12 11 27 54 27 25 45 27 10 29 9 i 25 12 41 55 55 27 56 14 28 11 Jl 20 27 26 13 12 24 56 28 26 43 29 11 23 31 54 27 13 42 53 57 28 57 11 3O 5 43 21 28 14 13 21 58 29 27 40 31 17 54 48 ' 29 14 43 50 59 29 58 8 32 1 6 14 ; 30 15 14 18 60 30 28 37 1 Lunation = 29 s 12" 44 m 3' 6 u 2l lv 14 V 24 V1 0"" In leap years, after February, a day and its motion must be added to the time for which the moon's mean distance from the sun is given. But when the mean time of any new or full moon is required in leap year after February, a day must be subtracted from the mean timethereof,as found >y the tables. In common years they give the day right. 87 A Table of the Sun's mean Motion from the Moon's ascending node. Years Years Years be- Com- Sun from node. of the of the fore and Sun from node plete Julian World after Christ years. J > II Period. O " 706 4006 7 6 17 9 11 7 2 3 56 714 8 4OOO Oil 4 55 12 7 22 11 39 1714 1008 3OOO 9 10 35 IT 13 8 11 17 2 2/14 2008 .4 20OO 6 10 5 28 14 9 O 22 25 3714 3008 IS lOOO 3 9 35 44 15 9 19 27 49 3814 3108 3 900 7 24 32 46 16 1O 9 35 31 3914 3208 <~> 800 9 29 48 17 10 28 40 55 4014 3308 ^ 700 4 24 26 -49 18 11 17 46 18 4114 340S J5 600 9 9 23 51 19 O 6 51 43 4214 3508 , 500 1 24 2O 53 20 O 26 59 24 4314 3608 J8 i 6 9 17 54 40 1 23 58 49 4414 3708 300 u 10 24 14 56 do 2 2O 58 13 4514 38O8 1 20 3 9 11 58 80 3 17 57 37 4614 3908 *S 100 7 24 8 59 JOO 4 14 57 2 4/14 4008 f 1 0961 200 8 29 54 3 4814 4108 1 101 4 24 3 3 30O 1 14 51 5 4pl4 42Q8 2O1 re 9904 4OO 5 2p 48 7 5014 4308 301 1 23 57 6 500 1O 14 45 8 5114 4408 401 6 8 54 8 10OO 8 29 30 17 5214 4508 3. 501 O 23 51 9 2OOO 5 29 O 33 57U 5008 !* 1001 9 8 36 18 3000 2 28 30 50 6414 5708 1701 4 23 15 30 4OOO 11 28 1 6 6466 5/60 1753 i 28 19 Months. Sun from node. 6514 5808 1801 8 25 44 44 t ' " ' o J3 -g S n\ ~ - , Complete un from nde. Jan. 0000 . ~ m ** b --N " >% years. so' " Feb. 1 2 11 48 o5 - !? *ifi ~ -^ 1 19 5 23 Mar. 2 1 16 39 r> t- 1 w 2S-s G ^^5 2 1 8 10 47 April 3 3 28 27 62- 9 o .2 * G o VM n 1 27 16 10 May 4 4 37 5/ o 2r^t> o o? 4 2 17 23 53 June 5 6 49 45 vrf J3 w^ r ~ a ^ 5 3 6 29 16 July 6 7 59 14 tn - -S - u - C ** *^ ^ c 6 3 25 34 40 Aug. 7 9 11 i g^M-2 Png 7 4 14 4O 3 Sept. 8 12 22 49 r a "2 s r^i 8 5 4 47 46 Oct. 9 13 32 18 5 J? l b J.H 9 5 23 53 9 Nov. O 15 44 5 0.2 " S ^ w v2 = -r ^ 10 6 12 58 33 Dec. 1 16 53 34 .ssi.;* i H i his table agrees with the ld stile until the year 1753 ; and after that, with the new. A Table of the Sun's mean Motion from the Moon's Ascending Node. p Sun from node.!' Sun from node. ] Sun from node. *< on . ' i H . ' " M. i a in E AT ' // HI c it in mi 1 1 2 1Q s. II in mi w Th. III III, v 09 A. 3ft 44 3 --* T" O Q 3 6 57 J 2 36 31 1 20 31 4 o 4 9 16 2 5 12 32 1 23 7 5 5 11 36 3 7 48 33 1 25 43 6 O 6 13 54 4 1O 23 34 1 28 9 7 O 7 16 13 5 12 59 35 1 31 55 8 O 8 18 32 6 Q 15 35 36 1 33 31 9 O 9 20 51 7 18 11 37 1 36 6 10 O 10 23 1O 8 20 47 38 1 38 42 11 11 25 29 9 O 23 23 39 1 41 18 12 O 12 27 48 10 25 58 40 1 43 54 13 O 13 30 7 11 28 33 41 1 46 36 14 O 14 32 26 12 0319 42 1 49 5 15 15 34 15 13 33 45 43 1 51 41 16 16 37 4 14 36 21 44 1 54 17 17 O 17 39 23 15 38 57 45 1 56 53 18 O 18 41 41 16 41 32 46 i 59 29 19 O 19 44 17 44 8 47 225 20 O 20 46 19 18 46 44 48 2 4 41 21- O 21 48 38 19 O 49 20 49 2 7 17 22 22 50 57 20 51 56 50 2 9 53 23 23 53 16 21 6 54 32 51 2 12 29 24 i 24 55 35 22 O 57 8 52 2 15 5 25 O 25 57 54 23 59 43 53 2 J7 41 26 . 27 13 24 1 2 19 54 2 2O 17 27 28 2 32 25 1 4 55 55 2 22 53 28 29 4 51 26 1 7 31 56 2 25 29 29 1 7 10 27 i 10 7 57 2 28 4 30 1 1 9 29 28 1 12 43 58 2 3O 40 31 1 2 \1 48 29 1 15 9 59 2 33 16 32 i 1 3 14 47 30 1 J7 55 60 2 35 52 In leap years, after February, add one day and one day's motion to the time at which the sun's mean dis- tance from the ascending node is required. A SUPPLEMENT TO THE PRECEDING LECTURES, BY THE AUTHOR. MECHANICS. The Description of a new and safe Crane, which has four different powers, adapted to different weights. 1 -L HE common crane consists only of a large D^scriptk wheel and axle ; and the rope, by \vhich goods ra * e ncA are drawn up from ships, or let down from the quay to them, -winds or coils round the axle, as the axle is turned by men walking in the wheel. But, as these engines have nothing 1 Our author received a reward of fifty pounds for the invention of this crane, fiom the Society for the encourage- ment of Arts ; and a description of it was honoured with a place among the Transactions of the Royal Society of London, see vol. xlv, p. 42. EQ. 9O MECHANICS. to stop the weight from running down, if any of the men happen to trip or fall in the wheel, the weight descends, and turns the wheel rapid- ly backward, and tosses the men violently about within it ; which has produced melancholy in- stances, not only of limbs broke, but even of lives lost, by the ill-judged construction of cranes. And besides, they have but one power for all sorts of weights ; so that they generally spend as much time in raising a small weight as in raising a great one. * These imperfections and dangers induced me to think of a method for remedying them. And for that purpose, I contrived a crane with a pro- per stop to prevent the danger, and with differ- ent powers suited to different weights ; so that there might be as little loss of time as possible : arid also, that when heavy goods are let down into ships, the descent may be regular and deli- berate. This crane has four different powers : and, I believe, it might be built in a room eight feet in width : the gib being on the outside of the room. Three trundles, with different numbers of staves, are applied to the cogs of a horizontal wheel with an upright axle ; and the rope that draws up the weight coils round the axle. The wheel has ninety-six cogs, the largest trundle twenty-four staves, the next largest has twelve, and the smallest has six. So that the largest trundle makes four revolutions for one revolu- tion of the wheel : the next makes eight, and the smallest makes sixteen. A winch is occa- sionally put upon the axis of either of these trundles, for turning it ; the trundle being then used that gives a power best suited to the weight : MECHANICS. 9i and the handle of the winch describes a circle in every revolution equal to twice the circumfer- ence of the axle of the wheel. So that the length of the winch doubles the power gained by each trundle. As the power gained by any machine, or en- gine whatever, is, in direct proportion, as the velocity of the power is to the velocity of the weight ; the powers of this crane are easily esti- mated, and they are as follows. If the winch be put upon the axle of the larg- est trundle, and turned four times round, the wheel and axle will be turned once round : and the circle described by the power that turns the winch, being, in each revolution, double the cir- cumference of the axle, when the thickness of the rope is added thereto ; the power goes through eight times as much space as the weight rises through : and therefore (making some allowance for friction) a man will raise eight times as much weight by the crane as he would by his natural strength without it : the power, in this case, be- ing as eight to one, If the winch be put upon the axis of the next trundle, the power will be as sixteen to one, be- cause it moves sixteen times as fast as the weight moves. If the winch be put upon the axis of the small- est trundle, and turned round, the power will b'* as thirty-two to one. But if the weight should be too great, even for this power to raise, the power may be doubled by drawing up the weight by one of the parts of a double rope, going under a pulley in the move- able block, which is hooked to the weight below the arm of the gib ; and then the power will be as sixty- four to one. That is, a man could then 92 MECHANICS. raise sixty-four times as much weight by the crane as he could raise by his natural strength without it; because, for every inch that the weight rises, the working power will move through sixty- four inches. By hanging a block with two pullies to the arm of the gib, and having two pullies in the move- able block that rises with the weight, the rope being doubled over and under these pullies, the power of the crane will be as 1 28 to one. And so, by increasing the number of pullies, the power may be increased as much as you please : always remembering, that the larger the pullies are, the less is their friction. While the weight is drawing up, the ratch- teeth of a wheel slip round below a catch or click that falls successively into them, and so hinders the crane from turning backward, and detains the weight in any part of its ascent, if the man who works at the winch should acci- dentally happen to quit his hold, or choose to rest himself before the weight be quite drawn up. In order to let down the weight, a man pulls down one end of a lever of the second kind, which lifts the catch of the ratchet-wheel, and gives the weight liberty to descend. But, if the descent be too quick, he pulls the lever a little farther down, so as to make it rub against the cuter edge of a round wheel ; by v\ hich means he lets down the weight as slowly as he pleases : and, by pulling a little harder, he may stop the weight, if needful, in any part of its descent. If he accidentally quits hold of the lever, the catch immediately falls, and stops both the weight and the whole machine. This crane is represented in Plate I, where A is the great wheel, and $ its axle on which MECHANICS. $3 rope C winds. This rope goes over a pulley D in the end of the arm of the gib E, and draws up the weight F, as the winch G is turn- ed round. H is the largest trundle, / the next, and K is the axis of the smallest trundle, which is supposed to be hid from view by the upright supporter L. A trundle M is turned by the great wheel, and on the axis of this trundle is fixed the ratchet-wheel JV, into the teeth of which the catch falls. P is the lever, from which goes a rope QQ, over a pulley R to the catch ; one end of the rope being fixed to the lever, and the other end to the catch. S is an elastic bar of wood, one end of which is screw- ed to the floor : and, from the other end goes a rope (out of sight in the figure) to the further end of the lever, beyond the pin or axis on which it turns in the upright supporter T. The use of this bar is to keep up the lever from rub- bing against the edge of the wheel /, and to let the catch keep in the teeth of the ratchet-wheel : but a weight hung to the farther end of the lever would do full as well as the elastic bar ami rope. When the lever is pulled down, it lifts the catch out of the ratchet-wheel, by means of the rope QQ, and gives the weight .F liberty to des- cend : but if the lever P be pulled a little far- ther down than what is sufficient to lift the catch O out of the ratchet-wheel TV, it will rub against the edge of the wheel U, and thereby hinder the too quick descent of the weight ; and will quite stop the weight if pulled hard. And if the man who pulls the lever, should happen inad- vertently to let it go, the elastic bar will sudden- ly pull it up, and the catch will fall down and stop the machine. 94 MECHANICS. two upright rollers above the axis or upper gudgeon of the gib E : their use is to let the rope C bend upon them, as the gib is turn- ed to either side, in order to bring the weight over the place where it is intended to be let down. N. B. The rollers ought to be so placed, that if the rope C be stretched close by their utmost sides, the half thickness of the rope may be per- pendicularly over the centre of the upper gudgeon of the gib. For then, and in no other position of the rollers, the length of the rope between the pulley in the gib and the axle of the great wheel will be always the same, in all positions of the gib : and the gib will remain in any position to which it is turned. When either of the trundles is not turned by the winch in working the crane, it may be drawn off from the wheel, after the pin near the axis of the trundle is drawn out, and the thick piece of wood is raised a little behind the outward sup- porter of the axis of the trundle. But this is iiot material ; for, as the trundle has no friction on its axis but what is occasioned by its weight, it will be turned by the wheel without any sens- ible resistance in working the crane. -^h <>i V 1i :-'/>T 'i -rii djvi^ Imr Vv^V) -oq T A Pyrometer, that makes the expansion of metals by heal visible to the Jive-and-forty thousandth part of an inch. Description The upper surface of this machine is repre- pyr'omeTer. Sented b 7 Fi g' 1 f Plate IL ItS frame AB CD is made of mahogany wood, on which is a circle PLATE ii, divided into 36O equal parts ; and within that Flg ' I>Sup< circle is another, divided into eight equal parts. MECHANICS. 95 If the short bar E be pushed one inch forward (or toward the centre of the circle) the index e will be turned 125 times round the circle of 360 parts or degrees. As 125 times 36O is 45,OOO, it is evident, that if the bar E be moved only the 45,OOO rk part of an inch, the index will move one degree of the circle. But as in my pyrometer the circle is nine inches in diameter, the motion of the index is visible to half a de- gree, which answers to the ninety thousandth part of an inch in the motion or pushing of the short bar E. One end of a long bar of metal F is laid into a hollow place in a piece of iron G, which is fix- ed to the frame of the machine ; and the other nd of this bar is laid against the end of the short bar E, over the supporting cross bar HI: and, as the end /"of the long bar is placed close against the end of the short bar, it is plain, that if F expands, it will push forward, and. turn the index e. The machine stands on four short pillars, high enough from a table, to let a spirit-lamp be put on the table under the bar F; and when that is done, the heat of the flame of the lamp expands the bar, and turns the index. There are bars of different metals, as silver, brass, and iron, all of the same length as the bar F, for trying experiments on the different ex- pansions of different metals, by equal degrees of heat applied to them for equal lengths of time ; which may be measured by a pendulum, that swings seconds. Thus, Put on the brass bar F 9 and set the index to Method of the 36O th degree : then put the lighted lamp un- usin s ir - der the bar, and count the number of seconds in which the index goes round the plate, from 96 MECHANICS; 360 to 36O again ; and then blow out the lamp, and take away the bar. This done, put on an iron bar F where the brass one was before, and then set the index to the 360 th degree again. Light the lamp and put it under the iron bar, and let it remain just as many seconds as it did under the brass one ; and then blow it out, and you will see how many degrees the index has moved in the circle; and by that means you will know in what pro- portion the expansion of iron is to the expansion of brass ; which I find to be as 210 is to 360, or as seven is to twelve* By this method, the relative expansions of different metals may be found. The bars ought to be exactly of equal size ; and to have them so, they should be drawn, like wire, through a hole. When the lamp is blown out, you will see the index turn backward : which shews that the metal contracts as it cools. The inside of this pyrometer is constructed as follows. rig. a . In Fig. 2, A a is the short bar which moves between rollers ; and, on the side a it has fifteen teeth in an inch, which take into the leaves of a pinion B (twelve in number) on whose axis is the wheel C of 10O teeth, which take into the ten leaves of the pinion Z), on whose axis is the wheel E of 10O teeth, which take into the ten leaves of the pinion F, on the top of whose axis is the index above mentioned. Now, as the wheels C and E have J 00 teeth each ; and the pinions D and F have ten leaves each, it is plain, that if the wheel C turns once round, the pinion F and the index on its axis will torn 100 times round. But, as the first. MECHANICS. 97 pinion B has only twelve leaves, and the bar A a that turns it has fifteen teeth in an inch, which is twelve and a fourth part more ; one inch mo- don of the bar will cause the last pinion Fto turn a hundred times round, and a fourth part of a hundred over and above, which is twenty- five. So that if A a be pushed one inch, Fvrill be turned 125 times round. A silk thread b is tied to the axis of the pinion D, and wound several times round it ; and the other end of the thread is tied to a piece of slender watch-spring G, which is fixed into the stud H. So that as the bary expands, and pushes the bar A a forward, the thread winds round the axle, and draws out the spring : and as the bar contracts, the spring pulls back the thread, and turns the work the contrary way, which pushes back the short bar A a against the long bar f. This spring always keeps the teeth of the wheels in contact with the leaves of the pinions, and so prevents any shake in the teeth. In Fig. 1 , the eight divisions of the inner circle Fg- * are so many thousandth parts of an inch in the expansion or contraction of the bars ; which is just one thousandth part of an inch for each divi- sion moved over by the index. A water-mill^ invented by Dr. Barker^ that has neither wheel nor trundle. This machine is represented by Fig. 1 of Plate Barker's III, in which A is a pipe or channel that brings J^^ water to the upright tube B. The water runs Fig. i,s down the tube, and thence into the horizontal trunk C, and runs out through holes at d and f Vol. II G $8 MECHANICS. near the ends of the trunk on the contrary sides thereof. The upright spindle D is fixed in the bottom of the trunk, and screwed to it below by the nut g; and is fixed into the trunk by two cross bars at/: so that, if the tube B and trunk C be turned round, the spindle D will be turned also. The top of the spindle goes square into the rynd of the upper mill-stone H 9 as in common mills ; and, as the trunk, tube, and spindle, turn round, the mill-stone is turned round thereby. The lower, or quiescent, mill-stone is represented by / ; and K is the floor on which it rests, and wherein is the hole L for letting the meal run through, and fall down into a trough, which may be about M. The hoop or case that goes round the mill-stone rests on the floor AT, and supports the hopper, in the common way. The lower end of the spindle turns in a hole in the bridge- tree GF, which supports the mill-stone, tube, spindle, and trunk. This tree is moveable on a pin at A, and its other end is supported by an iron rod N fixed into it, the top of the rod go- ing through the fixed bracket 0, and having a screw nut o upon it, above the bracket. By turn- ing this nut forward or backward, the mill-stone is raised or lowered at pleasure. While the tube B is kept full of water from the pipe A, and the water continues to run out from the ends of the trunk ; the upper mill- stone H 9 together with the trunk, tube, and spindle, turns round. But, if the holes in the trunk were stopped, no motion would ensue ; even though the tube and trunk were full of water. For, If there were no hole in the trunk, the pressure MECHANICS. 99 of the water would be equal against all parts of its sides within. But, when the water has free egress through the holes, its pressure there is entirely removed : and the pressure against the parts of the sides which are opposite to the holes, turns the machine.* * See Appendix for farther information on the consti uc- tion of Dr. Barker's mill. ED. G 2 HYDROSTATICS. A machine for demonstrating that, on equal bottoms, the pressure of fluids is in proportion to their perpendicular heights, without any regard to their quantities. i- 1 HIS is termed the Hydrostatical Paradox: anc ^ t ^ le macnme f r shewing it is represented in in Fig- 2 of Pl ate HI* I n which A is a box that Fig. 2, sup.' holds about a pound of water, abcde a glass- tube fixed in the top of the box, having a small wire within it ; one end of the wire being hook- ed to the end F of the beam of a balance, and the other end of the wire fixed to a moveable bottom, on which the water lies, within the box ; the bottom and wire being of equal weight with an empty scale (out of sight in the figure) hang- ing at the other end of the balance. If this scale be pulled down, the bottom will be drawn up within the box, and that motion will cause the water to rise in the glass-tube. Put one pound weight into the scale, which will move the bottom a little, and cause the water to appear just in the lower end of the tube at a ; which shews that the water pressexS with the force of one pound on the bottom ; put another pound into the scale, and the water will rise from a to b in the tube, just twice as high above the bottom as it was when at a; and then, as its pressure on the bottom supports two pound weight in the scale, it is plain that the pressure on the bottom is then ecuial to two HYDROSTATICS. 101 pounds. Put a third pound weight in the scale, and the water will be raised from b to c in the tube, three times as high above the bottom as when it began to appear in the tube at a; which shews, that the same quantity of water that pressed, but with the force of one pound on the bottom, when raised no higher than a, presses with the force of three pounds on the bottom when raised three times as high to c in the tube. Put a fourth pound weight into the scale, and it will cause the water to rise in the tube from c to d, four times as high as when it was all con- tained in the box, which shews that its pressure then upon the bottom is four times as great as when it lay all within the box. Put a fifth pound weight into the scale, and the water will rise in the tube from d to e, five times as high as it was above the bottom, before it rose in the tube ; which shews that its pressure on the bottom is then equal to five pounds, seeing that it supports so much weight in the scale. And so on, if the tube was still longer j for it would still require an additional pound put into the scale, to raise the water in the tube to an addi- tional height equal to the space de ; even if the bore of the tube was so small as only to let the wire move freely within it, and leave room for any water to get round the .wire. Hence we infer, that if a long narrow pipe or tube was fixed in the top of a cask full of liquor, and if as much liquor was poured into the tube as would fill it, even though it were so small as not to hold an ounce weight of liquor ; the pressure arising from the liquor in the tube would be as great upon the bottom, and be in as much danger of bursting it out, as if the cask was continued up, in its full size, 03 102 HYDROSTATICS. to the height of the tube, and filled with li- quor. Solution of In order to account for this surprising affair, dol 1 *" we must cons ider tnat fluids press equally in all manner of directions ; and consequently that they press just as strongly upward as they do downward. For, if another tube, as f, be put into a hole made into the top of the box, and the box be filled with water ; and then, if water be poured in at the top of the tube abcde, it will rise in the tube f to the same height as it does in the other tube : and if you leave off pouring, when the water is at c, or any other place in the tube abcde, you will find it just as high in the tube f: and if you pour in water to fill the first tube, the second will be filled also. Now, it is evident, that the water rises in the tube f, from the downward pressure of the wa- ter in the tube abcd^, on the surface of the water, contiguous to the inside of the top of the box ; and as it will stand at equal heights in both tubes, the upward pressure in the tube f is equal to the downward pressure in the other tube. But, if the tube/" were put in any other part of the top of the box, the rising of the water in it would still be the same : or, if the top was full of holes, and a tube put into each i of them, the water would rise as high in each tube as it was poured into the tube abode ; . and then the moveable bottom would have the weight of the water in all the tubes to bear, be- side the weight of all the water in the box. And seeing that the water is pressed upward into each tube, it is evident that, if they be all taken away, excepting the tube a b cd e, and the holes in which they stood be stopped up ; each part, thus stopped, will be pressed as much up^ HYDROSTATICS. 103 ward, as was equal to the weight of water in each tube. So that, the upward pressure against the inside of the top of the box, on every part equal in breadth to the width of the tube abcde, will be pressed upward with a force equal to the whole weight of water in the tube. And conse- quently, the whole upward pressure against the top of the box, arising from the weight or down- ward pressure of the water in the tube, will be equal to the weight of a column of water of the same height with that in the tube, and of the same thickness as the width of the inside of the box : and this upward pressure against the top will re-act downward against the bottom, and be as great thereon, as would be equal to the weight of a column of water as thick as the moveable bottom is broad, and as high as the water stands in the tube. And thus, the para- dox is solved. The moveable bottom has no friction against the inside of the box, nor can any water get between it and the box. The method of making it so, is as follows. In Fig. 3, A BCD represents a section of the Construe- box, and abed is the lid or top thereof, which tion of ' h ~ . , ... i i- i r moveable goes on tight, like the lid or a common paper bottom, snuff-box. R is the moveable bottom, with a groove around its edge, and it is put into a bladder fg, which is tied close around it in the Fig. 3. groove by a strong waxed thread ; the bladder coming up like a purse within the box, and put over the top of it at a and d all round, and then the lid pressed on. So that, if water be poured in through the hole // of the lid, it will lie upon the Tx>ttom E, and be contained in the space fEgh within the bladder ; and the bottom may be raised by pulling the wire i, which is fixed to 104 HYDROSTATICS. it at E : and by thus pulling the wire, the water will be lifted up in the tube k, and as the bot- tom does not touch the inside of the box, it moves without friction. Now, suppose the diameter of this round bot- tom to be three inches (in which case, the area thereof will be nine circular inches), and the diameter of the bore of the tube to be a quarter of an inch ; the whole area of the bottom will be J 44 times as gre,at as the area of the top of a pin that would fill the tube like a cork. And hence it is plain, that if the moveable bottom be raised only the 144 th part of an inch, the water will thereby be raised a whole inch in the tube ; and consequently, that if the bottom be raised one inch, it would raise the water to the top of a tube 144 inches, or twelve feet in height. JV. B. The box must be open below the move- able bottom, to let in the air. Otherwise, the pressure of the atmosphere would be so great upon the moveable bottom, if it be three inches in diameter, as to require 108 pounds in the scale, to balance that pressure, before the bot- tom could begin to move. A machine^ to be substituted in place of the com- mon hydrostatical bellows. Substitute IN Fig. 1 of Plate IV, AE CD is an oblong for Uie hy- S q Uare box, in one end of which is a round drostatical r , r bellows, groove, as at a, rrom top to bottom, tor receiv- PLATE iv, ing the upright glass tube /, which is bent to a Kg- 1, a, 3, right angle at the lower end (as at i in Fig. 2), < Sup ' and to that part is tied the neck of a large blad- der K (Fig. 2), which lies in the bottom of the HYDROSTATICS. 105 box. Over this bladder is laid the moveable board L (Fig. 1 and 3), in which is fixed an upright wire M ; and leaden weights NN, to the amount of sixteen pounds, with holes in their middle, which are put upon the wire, over the board, and press upon it with all their force. The cross bar p is then put on, to secure the tube from falling, and keep it in an upright po- sition : and then the piece EFG is to be put on, the part G sliding tight into the dove-tailed groove H, to keep the weights NN horizontal, and the wire M upright ; there being a round hole e in the part E F for receiving the wire. There are four upright pins in the four cor- ners of the box within, each almost an inch long, for the board L to rest upon : to keep it from pressing the sides of the bladder below it close together at first. The whole machine being thus put together, pour water into the tube at top ; and the water will run down the tube into the bladder below the board ; and after the bladder has been filled up to the board, continue pouring water into the tube, and the upward pressure which it will excite in the bladder, will raise the board with all the weight upon it, even though the bore of the tube should be so small, that less than an ounce of water would fill it. ' 1 Upon this principle, it has been justly affirmed by some writers on natural philosophy, that a certain quantity of water, however small, may be rendered capable of ex- erting a force equal to any assignable one, by increasing the height of the column, and diminishing the base on which it presses. Dr. Goldsmith observes, that he has seen a strong hogshead split in this manner. A small, though strong tube of tin, twenty feet high, was inserted iu 1O6 HYDROSTATICS. This machine acts upon the same principle as the one last described, concerning the Hydrosta- tical paradox. For, the upward pressure against every part of the board (which the bladder touches), equal in area to the area of the bore of the tube, will be pressed upward with a force equal to the weight of the water in the tube ; and the sum of all these pressures against so many areas of the board, will be sufficient to raise it with all the weights upon it. In my opinion, nothing can exceed this simple machine, in making the upward pressure of fluids evident to sight. The cause of reciprocating springs, and of ebbing- and flowing wells, explained? IN Fig. 1 of Plate V, let abed be a hill, gs .within which is a large cavern A A near the top, PLATE v fiH ec l or && by rains and melted snow on the Fig. i, sup. top a, making their way through chinks and in the bung-hole of the hogshead. Water was then poured into the tube till the hogshead was filled, and the water had reached within a foot of the top of the tin tube. By the pressure of this column of water, the hogshead burst with incredible force, and the water was scattered in every direction By diminishing the area of the tube one half, or doubling its height, the same quantity of water would have a double force. ED. * Dr. Atwell of Oxford seems to have been the first person that pointed out the cause of reciprocating springs. The theory of this gentleman, of which the article in the text is an abridgement, was published in Number 424 of the hilosophical Transactions, and was suggested by the phenomena of Laywell spring, at Br'txam in Devonshire. >ee Desagulier's Experimental Philosophy, vol. ii, p. 173. and vol. i, of this work, p. 141. ED. 3 HYDROSTATICS. 107 crannies into the said cavern, from which pro- ceeds a small stream CC within the body of the hill, and issues out in a spring at G on the side of the hill, which will run constantly while the cavern is fed with water. From the same cavern *AA^ let there be a small channel /), to carry water into the cavern B ; and from that cavern let there be a bended channel E e F, larger than _D, joining with the former channel CC, as at f before it comes to the side of the hill ; and let the joining at f be below the level of the bottom of both these ca- verns. As the water rises in the cavern J3, it will rise as high in the channel EeF: and when it rises to the top of that channel at e 9 it will run down the part c FG, and make a swell in the spring G, which will continue till all the water is drawn off from the cavern , by the natural syphon EeF (which carries off the water faster from B than the channel D brings water to it), and then the swell will stop, and only the small channel CC will carry water to the spring G, till the ca- vern B is filled to B again by the rill D ; and then the water being at the top e of the channel EeF, that channel will act again as a syphon, and carry off all the water from B to the spring G, and so make a swelling flow of water at G as before. To illustrate this by a machine (Tig. 2), let ./^illustrated be a large wooden box, filled with water; andJJ^*" let a small pipe CC (the upper end of which is fixed into the bottom of the box) carry water from the box to G, rf where it will run off con- % *- stantly, like a small spring. Let another small pipe D carry water from the same box to the box or well B, from which let a syphon E e F 108 HYDROSTATICS. proceed, and join with the pipe CC at f: the bore of the syphon being larger than the bore of the feeding- pipe D, As the water from this pipe rises in the well .5, it will also rise as high in the syphon EeF ; and when the syphon is full to the top e, the water will run over the bend e, down the part eF, and go off at the mouth G ; which will make a great stream at G : and that stream will continue, till the sy- phon has carried off all the water from the well B ; the syphon carrying off the water faster from B than the pipe D brings water to it : and then the swell at G will cease, and only the water from the small pipe C C will run off at G, till the pipe D fills the well B again ; and then the syphon will run, and make a swell at G as before. And thus, we have an artificial representation of an ebbing and flowing well, and of a reci- procating spring, in a very natural and simple manner. HYDRAULICS. An account of the principles by which Mr. Blakt-y proposes to raise water from mines, or from rivers, to supply towns, and gentle?nen\s seats, by his new-invented fire-engine, for which he has received his majesty's letters patent. ALTHOUGH I am not at liberty to describe Biakey's . the whole of this simple engine, yet I have the T en s me - patentee's leave to describe such a one as will pjg. 4l S up. shew the principles by which it acts. In Fig. 4 of Plate IV, let A be a large, strong, close, vessel, immersed in water up to the cock b, and having a hole in the bottom, with a valve a upon it, opening upward within the vessel. A pipe B C rises from the bottom of this ves- sel, and has a cock c in it near the top, which is small there, for playing a very high jet d. K is the little boiler (not so big as a common tea-kettle) which is connected with the vessel A by the steam-pipe F ; and G is a funnel, through which a little water must be occasionally poured into the boiler, to yield a proper quantity of steam ; and a small quantity of water will do for that purpose, because steam possesses up- ward of 14,000 times as much space or bulk as the water does from which it proceeds. The vessel A being immersed in water up to the cock b, open that cock, and the water will rush in through the bottom of the vessel at a, and fill it as high up as the water stands on its outside ; and the water, coming into the vessel, 1IO HYDRAULICS. will drive the air out of it (as high as the water rises within it) through the cock b. When the water has done rushing into the vessel, shut the cock b, and the valve a will fall down, and hin- der the water from being pushed out that way, by any force that presses on its surface. All the part of the vessel above b will be full of common air when the water rises to b. Shut the cock c, and open the cocks d and e ; then pour as much water into the boiler E (through the funnel G) as will about half fill the boiler ; and then shut the cock d, and leave the cock e open. This done, make a fire under the boiler E, and the heat thereof will raise a steam from the water in the boiler ; and the steam will make its way thence, through the pipe JF, into the vessel A ; and the steam will compress the air (above Z>) with a very great force upon the sur- face of the water in A* When the top of the vessel A feels very hot by the steam under it, open the cock c in the pipe C ; and the air being strongly compressed in A, between the steam and the water therein, will drive all the water out of the vessel A, up the pipe -5C, from which it will fly up in a jet to a very great height. In my fountain, which is made in this manner after Mr. Blakey's, three tea-cup-fulls of water in the boiler will afford steam enough to play a jet thirty feet high. When all the water is out of the vessel A, and the compressed air begins to follow the jet, open the cocks b and d to let the steam out of the boiler E and vessel A^ and shut the cock e to prevent any more steam from getting into A ; and the air will rush into the vessel A through the cock , and the water through the valve a : HYDRAULICS. Ill und so the vessel will be filled with water, up to the cock b as before. Then shut the cock b 9 and the cocks c and scrcw - of the letters, by the fell of water EF, which engllc< need not be more than three feet. The axle G Fig A "su P ! of the wheel is elevated so as to make an angle of about 44 J with the horizon; and on the top of that axle is a wheel //, which turns such ano- ther wheel / of the same number of teeth : the axle K of this last wheel being parallel to the axle G of the two former wheels. The axle G is cut into a double-threaded screw (as in Fig. 2), exactly resembling the screw on Fig. %. the axis of the fly of a common jack, which must be (what is called) a right-handed screw, like the wood-screws, if the first wheel turns in the direction AE CD ; but must be a left-handed screw, if the stream turns the wheel the contrary way. And, whichever way the screw on the axle G be cut, the screw on the axle K must be cut the contrary way ; because these axles turn in contrary directions. The screws being thus cut, they must be covered close over with boards, like those of a cylindrical cask ; and then they will be spiral tubes. Or, they may be made of tubes of stift" leather, and wrapt round the axles in shallow grooves cut therein, as in Fig. 3. The lower end of the axle G turns constantly Fig. 3. in the stream that turns the wheel, and the lower ends of the spiral tubes are open into the water ; so that, as the wheel and axle are turned round, the water rises in the spiral tubes, and runs out at L, through the holes MN 9 as they come about below the axle. These holes (of which there Vol. II. II 114 HYDRAULICS. may be any number, as four or six) are in a, broad close ring on the top of the axle, into which ring the water is delivered from the up- per open ends of the screw-tubes, and falls into the open box j^. The lower end of the axle K turns on a gudgeon, in the water in N; and the spiral tubes in that axle take up the water from JV, and deliver it into such another box under the top of K ; on which there may be such another wheel as /, to turn a third axle by such a wheel upon it. And in this manner, water may be raised to any given height, when there is a stream sufficient for that purpose to act on the broad float-boards of the first wheel. 3 3 As Mr. Ferguson has not explained the reason why the water rises in the spirals of the screw engine, we hope the reader will understand it from the following remarks. . When the screw B F, in Figure 3, Plate VI, is in a ver- tical position, the spiral excavations will be inclined to the horizon, and if a portion of water be introduced at the top A, it will descend to F, the bottom of the tube. If the screw be in a horizontal position, and the water intro- duced at , it will fall to C, and remain there. But if the screw be turned upon its axis from B towards A, so that the lowest point C of the tube may ascend to D, while the point B is depressed to C, the water will, by its own gra- vity, move from C to , where it will be discharged : So that water introduced into one extremity of the screw engine, in a horizontal position, will be discharged at the other. Now, let the end JB, of the engine B F y be ele- vated so as to be inclined to the horizon, and the same effect will be produced : the water at C will rise towards jB, till the angle of inclination which the machine make with the horizon is equal to the angle formed by the spirals with the axis of the engine. At this particular angle the water will have as great a tendency to flow to- wards D as towards , because the surface of the tube between these two points is parallel to the horizon -, but at HYDRAULICS. 115 A quadruple pump-mill for raising water. This engine is represented on Plate VII, which ABCD is a wheel, turned by water ac- pump ~ n cording to the order of the letters. On the ho- rizontal axis are four small wheels, toothed al- most half round : and the parts of their edges on which there are no teeth are cut down so, as to be even with the bottoms of the teeth where they stand. The teeth of these four wheels take alternate- ly into the teeth of four racks, which hang by two chains over the pulleys Q and L ; and to the lower ends of these racks there are four iron rods fixed, which go down into the four forcing pumps, S, R 9 My and N. And, as the wheels turn, the rack and pump-rods are alternately moved up and down. Thus, suppose the wheel G has pulled down at a greater angle, the fluid will descend towards D t and flow out at the extremity F- The ascension of the water, therefore, in the Archimedean screw engine arises from its tendency to occupy the lowest parts of the spiral, while the rotatory motion withdraws this part of the spiral from the fluid, and causes it to ascend to the top of the tube. By wrapping a right angled triangle round a cylindrical pin, so that the hypothenuse may form a spiral upon its surface, and by attending to the position of the spirals at different angles of inclination, the preceding observations will be easily understood. In practice, the angle of inclination should be about 50, and the angle which the spirals form with the axis should exceed the angle of the engine's incline lion by about ] 5. The theory of this engine is treated at great length by Hennert, in his Distcrtation sur la vu jy Arch'imede^ Berlin, 1767 ; and by Euler, in the Nov. Comment. Petrop. torn. v. See also Gregory's Mechanics, voL ii, p. 343. ED. H2 116 HYDRAULICS. the rack /, and drawn up the rack K by the chain ; as the last tooth of G just leaves the uppermost tooth of 7, the first tooth of H is ready to take into the lowermost tooth of the rack K, and pull it down as far as the teeth go ; and then the rack / is pulled upward through the whole space of its teeth, and the wheel G is ready to take hold of it, and pull it down again, and so draw up the other. In the same manner, the wheels E and F work the racks and P. * These four wheels are fixed on the axle of the great wheel in such a manner, with respect to the positions of their teeth, that while they con- tinue turning round, there is never one instant of time in which one or other of the pump-rods is not going down, and forcing the water. So that, in this engine, there is no occasion for having a general air-vessel to all the pumps, to procure a constant stream of water flowing from the upper end of the main pipe. The pistons of these pumps are solid plungers,^ the same as described in Lecture fifth, volume first. See Plate XI, Fig. 4 5 with the description of the figure. From each of these pumps, near the lowest end, in the water, there goes off a pipe, with a valve on its farthest end from the pump ; and these ends of the pipes all enter one close box, into which they deliver the water : and into this box, the lower end of the main conduit-pipe is * For the proper form which must be given to the teeth of the wheels and racks, in order to produce an equable and uniform motion, see Appendix. This method of moving the pistons is preferable to the crank motion em- ployed in the engine which is represented in Plate XIT, Vol. i. ED. HYDRAULICS. 11? fixed ; so that, as the water is forced or pushed into this box, it is also pushed up the main pipe to the height that it is intended to be raised. There is an engine of this sort, described in Ramelli's work : but I can truly say, that I never saw it till some time after I had made this model. The said model is not above twice as big as the figure of it, here described, r rurn it by a winch fixed on the gudgeon of the axle behind the water wheel ; and when it was newly made, and the pistons had valves in good order, I put tin pipes 15 feet high upon it, when they were joined together, to see what it could do ; and I found, that in turning it moderately by the winch, it would raise a hogshead of water in an hour, to the height of 15 feet. H 3 DIALING. The dniverscd Dialing Cylinder. Universal IN Fig. 1, of Plate VIII, A BCD represents a finder 8 cy ~ cylindrical glass tube, closed at both ends with Pi ATE brass plates, and having a wire or axis EFG vin, fixed in the centres of the brass plates at top Fig.i,Sup. an( j bottom. This tube is fixed to a horizontal board H 9 and its axis makes an angle with the board equal to the angle of the earth's axis with the horizon of any given place, for which the cylinder is to serve as a dial. And it must be set with its axis parallel to the axis of the world in that place ; the end E pointing to the ele- vated pole. Or, it may be made to move upon a joint ; and then it may be elevated for any par- ticular latitude. There are 24 straight lines, drawn with a dia- mond, on the outside of the glass, equi-distant from eaeh other, and all of them parallel to the axis. These are the hour-lines ; and the hours are set to them as in the figure : the XII next B stands for midnight, and the opposite XII, next the board H 9 stands for mid-day or noon. The axis being elevated to the latitude of tne place, and the foot-board set truly level, with the black line along its middle in the plane of the meridian, and the end JV toward the north j the axis EFG will serve as a stile or gnomon, and DIALING. 119 cast a shadow on the hour of the day, among the parallel hour-lines when the sun shines on the machine, For, as the sun's apparent diurnal motion is equable in the heavens, the shadow of the axis will move equably in the tube ; and will always fall upon that hour-line which is opposite to the sun, at any given time. The brass plate A D, at the top, is parallel to the equator, and the axis EFG is perpendicular to it. If right lines be drawn from the centre of this plate to the upper ends of the equi-distant parallel lines on the outside of the tube ; these right lines will be the hour-lines on the equi- noctial dial AD) at 15 distance from each other : and the hour letters may be set to them, as in the figure. Then, as the shadow of the axis within the tube comes on the hour-lines of that tube, it will cover the like hour-lines on the equinoctial plate AD. If a thin horizontal plate ef be put within the tube, so as its edge may touch the tube all around ; and right lines be drawn from the cen- tre of the plate to these points of its edge which are cut by the parallel hour-lines on the tube ; these right-lines will be the hour-lines of a hori- zontal dial, for the latitude to which the tube is elevated. For, as the shadow of the axis comes successively to the hour-lines of the tube, and covers them, it will then cover the like hour- lines on the horizontal plate ef, to which the hours may be set, as in the figure. If a thin vertical plate g C, be put within the tube, so as to front the meridian, or 1 2 o'clock line, thereof, and the edge of this plate touch the tube all around : and then, if right lines be drawn from the centre of the plate to those points of its edge which are cut by the parallel hour- 120 IMALINO. lines on the tube ; these right lines will be th* hour-lines qf a vertical south dial ; and the sha- dow of the axis will cover them at the same times when it covers those of the tube. If a thin plate be put within the tube so as to decline, or incline, or recline, by any given num- ber of degrees ; and right lines be drawn from its centre to the hour-lines of the tube ; these right lines will be the hour-lines of a declining, inclining, or reclining, dial, answering to the like number of degrees, for the latitude to which the tube is elevated. And thus, by this simple-machine, all the prin- ciples of dialing are made very plain and evi- dent to the sight. And the axis of the tube (which is parallel to the axis of the world in every latitude to which it is elevated) is the stile or gnomon for all the different kinds of sun-dials. And, lastly, if the axis of the tube be drawn out, with the plates AD, ef, and g C upon it ; and set it up in sun-shine, in the same position as they were in the tube ; you will have an equi- noctial dial AD, a horizontal dial ef, and a ver- tical south dial gC; on all which the time of the day will be shewn by the shadow of the axis or gnomon EFG. Let us now suppose that, instead of a glass tube, AB CD is a cylinder of wood, on which the 24 parallel hour lines are drawn all around, at equal distances from each other ; and that, from the points at top, where these lines end, right lines are drawn toward the centre, on the flat surface AD : these right lines will be the hour-lines on an equinoctial dial, for the latitude of the place to which the cylinder is elevated above the horizontal foot or pedestal H; and they are equidistant from each other, as in Fig. 2; DIALING. 121 which is a full view of the flat surface or top pg. a. AD of the cylinder, seen obliquely in Fig. 1. And the axis of the cylinder (which is a straight wire E F G all down its middle) is the stile or gnomon, which is perpendicular to the plane of the equinoctial dial, as the earth's axis is per- pendicular to the plane of the equator. To make a horizontal dial, by the cylinder, for any latitude to which its axis is elevated ; draw out the axis and cut the cylinder quite through, as at e hfg, parallel to the horizontal board H, and take off the top part eADfe ; and the section e hfg e will be of an elliptical form, as in Fig. 3. Then, from the points of this Fig. 3, section (on the remaining part eCf), where the parallel lines on the outside of the cylinder meet it, draw right lines to the centre of the section ; and they will be the true hour-lines for a horizontal dial, as ale da in Fig. 3, which may be included in a circle drawn on that sec- tion. Then put the wire into its place again, and it will be a stile or casting a shadow on the time of the day, on that dial. So E (Fig. 3) is the stile of the horizontal dial, parallel to the axis of the cylinder. To make a vertical south dial by the cylinder, draw out the axis, and cut the cylinder perpen- dicularly to the horizontal board H, as at giCkg, beginning at the hour-line (B geA) of XII, and making the section at right angles to the line SHN on the horizontal board. Then, take off the upper part gADC t and the face of the section thereon will be elliptical, as shewn in Fig. 4. From the points in the edge of this Fig. 4. section, where the parallel hour-lines on the round surface of the cylinder meet it, draw right line* to the centre of the section ; and they will 122 DIALING. be the true hour-lines on a vertical direct south dial, for the latitude to which the cylinder was elevated ; and will appear as in Fig. 4, on which the vertical dial may be made of a circular shape, or of a square shape, as represented in the figure ; and F will be its stile parallel to the axis of the cylinder. And thus, by cutting the cylinder any way, so as its section may either incline, or decline, or recline, by any given number of degrees ; and from those points in the edge of the section, xvhere the outside parallel hour-lines meet it, draw right lines to the centre of the section ; and they will be the true hour-lines for the like de- clining, reclining, or inclining, dial : and the axis of the cylinder will always be the gnomon or stile of the dial ; for, whichever way the plane of the dial lies, its stile (or the edge thereof that casts the shadow on the hours of the day) must be parallel to the earth's axis, and point toward the elevated pole of the heaven. To delineate a. sun-dial on paper, which, when pasted round a cylinder of wood, shall shew the time of the day, tlie sun's place in the ecliptic^ and his altitude, at any time of observation, See Plate IX. Sup. PLATE ix, Draw the right line aAE, parallel to the top Su P' of the paper ; and with any convenient opening of the compasses set one foot in the end of the line at a, as a centre, and with the other foot describe the quadrantal arc A E, and divide it into 90 equal parts or degrees. Draw the right line AC, at right angles to aAE, and touching the quadrant A E at the point A. Then, from the DIALING. centre a, draw right lines through as many de- grees of the quadrant as are equal to the sun's altitude at noon, on the longest day of the year, at the place for which the dial is to serve ; which altitude at London is 62 degrees : and continue these right lines till they meet the tangent line AC, and from these points of meeting, draw straight lines across the paper, parallel to the first right line AB^ and they will be the paral- lels of the sun*s altitude, in whole degrees, from sun-rise till sun-set, on all the days of the year. These parallels of altitude must be drawn out to the right line D, which must be parallel to A C, and as far from it as is equal to the intend- ed circumference of the cylinder on which the paper is to be pasted, when the dial is drawn upon it. Divide the space between the right lines A C and B D (at top and bottom) into twelve equal parts, for the twelve signs of the ecliptic ; and, from mark to mark of these divisions at top and bottom, draw right lines parallel to AC and B D; and place the characters of the 12 signs in these twelve spaces, at the bottom, as in the figure : beginning with vy or Capricorn, and ending with x or Pis.ces. The spaces including the signs should be divided by parallel lines into halves ; and if the breadth will admit of it with- out confusion, into quarters also. At the top of the dial, make a scale of the months and days of the year, so as the days may stand over the sun's place for each of them in the signs of the ecliptic. The sun's place, for every day of the year, may be found by any common ephemeris : and here it will be best to make use of an ephemeris for the second year after leap-year ; as the nearest means for the svm's 124 DIALING* place on the days of the leap-year, and on those of the first, second, and third year after. Compute the sun's altitude for every hour (in the latitude of your place), when he is in the beginning, middle, and end, of each sign of the ecliptic ; his altitude at the end of each sign being the same as at the beginning of the next. And, in the upright parallel lines, at the begin- ning and middle of each sign, make marks for those computed altitudes among the horizontal parallels of altitude, reckoning them downward, according to the order of the numeral figures set to them at the right hand, answering to the like division of the quadrant at the left ; and, through these marks, draw the curve hour-lines, and set the hours to them, as in the figure, reckoning the forenoon hours downward, and the afternoon hours upward. The sun's altitude should also be computed for the half hours ; and the quarter-lines may be drawn, very nearly in their proper places, by estimation and accuracy of the eye. Then, cut off the paper at the left hand, on which the quadrant was drawn, close by the right line A C 1 , and all the paper at the right hand close by the right line B D ; and cut it also close by the top and bottom horizontal lines ; and it will be fit for pasting round the cylinder. x, This cylinder is represented in miniature by F * J > Su F-Fig. 1, Plate X. It should be hollow, to hold the stile D E when it is not used. The crooked end of the stile is put into a hole in the top AD of the cylinder ; and the top goes on tightish, but must be made to turn round on the cylinder, like the lid of a paper snuff box. The stile must stand straight out, perpendicular to the side of the cylinder, just over the right line AB iii DIALING. 125 Plate IX, where the parallels of the sun's alti- PLATE ix, tude begin : and the length of the stile, or dis- Sup ' tance of its point e from the cylinder, must be equal to the radius a A of the quadrant A E in Plate IX. The method of using this dial is as follow.- Place the horizontal foot B C of the cylinder PL AT x, on a level table where the sun shines, and turn Jg *' the top AD till the stile stands just over the day of the then present month. Then turn the cylinder about on the table, till the shadow of the stile falls upon it, parallel to those upright lines, which divide the signs, that is, till the shadow be parallel to a supposed axis in the middle of the cylinder : and then, the point, or lowest, end of the shadow, will fall upon the time of the day, as it is before or after noon, among the curve hour-lines ; and will shew the sun's altitude at that time, among the cross pa- rallels of his altitude, which go round the cylin- der : and, at the same time, it will shew in what sign of the ecliptic the sun then is, and you may- very nearly guess at the degree of the sign, by estimation of the eye. The ninth plate, on which this dial is drawn, may be cut out of the book, and pasted round a cylinder whose length is 6 inches and 6 tenths of an inch below the moveable top ; and its diameter '2 inches and 24 hundred parts of an inch. Or, I suppose the copper-plate prints of it may be had of the booksellers in London. But it will only do for London, and other places of the game latitude. a level table cannot be had, the dial I 1 26 DIALING. may be hung by the ring F at the top ; and when it is not used, the wire that serves for a stile may be drawn out, and put up within the cylinder ; and the machine carried in the pocket. To make three Sun-dials upon three different planes, so as they may all shew the time of the day by one gnomon. On the flat board A B C, describe a horizontal 'dial, according to any of the rules laid down in the Lecture on Dialing ; and to it fix its gnomon FG //, the edge of the shadow from the side FG being that which shews the time of the day. To this horizontal or flat board, join the up- right board ED (?, touching the edge G H of the gnomon. Then, making the top of the gnomon at G the centre of the vertical south dial, describe a south dial on the board E D C. Lastly, on a circular plate IK describe an equinoctial dial, all the hours of which dial are equidistant from each other ; and making a slit c d in that dial, from its edge to its centre, in the XII o'clock line, put the said dial perpendicu- larly on the gnomon FG, as far as the slit will admit of; and the triple dial will be finished ; the same gnomon serving all the three, and shewing the same time of the day on each of them, 1 1 This dial may be converted into a portable and uni- versal one by a very simple contrivance. Remove the stile F HG and the dial K, and make EDC turn upon H C as a hinge, so that it may fold down upon A B, and thus go into very small compass when not used. Fix a silk thread at F, and having divided the line G H continued, into a line of tangents for the radius FH, make a small holr DIALING* 127 An universal Dial on a plain cross. This dial is represented by Fig. 1, of Plate universal XI, and is moveable on a joint C, for elevating PLATE xi, it to any given latitude, on the quadrant C O 9O, Fi s- '* SU P- as it stands upon the horizontal board A. The arms of the cross stand at right angles to the middle part ; and the top of it from a to n, is of equal length with either of the arms n e or m k. Having set the middle line t u to the latitude of your place, on the quadrant, the board A level, and the point N northward by the needle ; the plane of the cross will be parallel to the plane of the equator ; and the machine will be rectified. Then, from III o'clock in the morning, till VI, the upper edge k I of the arm i o will cast a shadow on the time of the day on the side of the arm cm: from VI till IX the lower edge i of the arm / o will cast a shadow on the hours on the side o q. From IX in the morning till XII at noon, the edge a b of the top part a n will cast a shadow on the hours on the arm n ef : from XII to III in the afternoon, the edge c d of the top part will cast a shadow on the hours on the hole through the board at every degree of the line of tan- gents. Extend the silk thread from F towards G, making it pass through the hole at the degree of the line of tan- gents answering to the latitude of the place. The thread will 'then be the gnomon of the horizontal dial ABC, which is set due south, by means of a small mariner's com- pass placed between F and ff, allowance being made for the variation. The vertical south dial C serves only for a place, the tangent of whose latitude is equal to H G. This dial is not altogether correct, but is remarkably con- vaeient for carrying in the pocket. ED. 128 DIALING. arm him : from III to VI in the evening the edge g h will cast a shadow on the hours on the part p s ; and from VI till IX, the shadow of the edge ef will shew the time on the top part a n. The breadth of each part a , e/, &c. must be so great as never to let the shadow fall quite without the part or arm on which the hours are marked, when the sun is at his greatest declina- tion from the equator. To determine the breadth of the sides of the arms which contain the hours, so as to be in just proportion to their length, make an angle ABC Fig. *. (Fig. 2) of 23^-, which is equal to the sun's greatest declination : and suppose the length of each arm, from the side of the long middle part, and also the length of the top part above the arms, to be equal to B d. Then, as the edges of the shadow from each of the arms, will be parallel to B o, making an angle of 23^- with the side B n of the arm when the sun's declination is 23y ; it is plain, that if the length of the arm be B n, the least breadth that it can have, to keep the edge B o of the shadow B o g d from going off the side of the arm n o before it comes to the end o n thereof, must be equal to on or d B. But in order to keep the shadow within the quar- ter divisions of the hours, when it comes near the end of the arm, the breadth thereof should be still greater, so as to be almost doubled, on account of the distance between the tips of -the arms. To place the hours right on the arms, take the following method. Fig. 3. Lay down the cross abed (Fig. 3) on a sheet of paper j and with a black lead pencil, held DIALING. 129 close to it, draw its shape and size on the paper. Then, taking the length a e in your compasses, and setting one foot in the corner a, with the other foot describe the quadrantal arc ef. Di- vide this arc into six equal parts, and through the division marks draw right lines a g, a h, &c. continuing three of them to the arm c e, which are all that can fall upon it ; and they will meet the arm in these points through which the lines that divide the hours from each other (as in Fig, 1) are to be drawn right across it. Divide each arm, for the three hours it con- tains in the same manner ; and set the hours to their proper places (on the sides of the arms), as they are marked in Fig. 3. Each of the hour spaces should be divided into four equal parts, for the half hours and quarters, in the quadrant cf; and right lines should be drawn through these division marks in the quadrant, to the arms of the cross, in order to determine the places thereon where the sub-divisions of the hours must be marked. This is a very simple kind of universal dial ; it is very easily made, and will have a pretty un- common appearance in a garden. I have seen a dial of this sort, but never saw one of the kind that follows. Vol. II. ISO DIALING. An universal Dial, shewing the hours of the day by a terrestrial globe, and by the shadows of several gnomons at the same time : together with all the places of the earth which are then enlightened by the sun ; and those to which the sun is then rising, or on the meridian, or set- ting. This dial (See Plate XII,) is made of a thick terrestrial sc l uare piece of wood, or hollow metal. The gfobe and sides are cut into semicircular hollows, in which several fa Q hpurs are placed ; the stile of each hollow gnomons. . y r ' PLATE commg out from the bottom thereof, as tar as xn, sup. t h e en j s o t he hollows project. The corners are cut out into angles, in the insides of which, the hours are also marked ; and the edge of the end of each side of the angle serves as a stile for casting a shadow on the hours marked on the other side. In the middle of the uppermost side or plane, there is an equinoctial dial : in the centre where- of an upright wire is fixed, for casting a shadow on the hours of that dial, and supporting a small terrestrial globe on its top. The whole dial stands on a pillar, in the middle of a round horizontal board, in which there is a compass and magnetic needle, for placing the meridian stile toward the south. The pillar has a joint with a quadrant upon it, divided into QO degrees (supposed to be hid from sight under the dial in the figure), for setting it to the latitude of any given place, the same way as already describ- ed in the dial on the cross. The equator of the globe is divided into 24 equal parts, and the hours are laid down upon DIALING. 131 it at these parts. The time of the day may be known by these hours, when the sun shines upon the globe. To rectify and use this dial, set it on a level table, or sole of a window, where the sun shines, placing the meridian stile due south, by means of the needle ; which will be, when the needle points as far from the north fleur-de-lis toward the west, as it declines westward, at your place.* Then bend the pillar in the joint, till the black line on the pillar comes to the latitude of your place in the quadrant. The machine being thus rectified, the plane of its dial-part will be parallel to the equator, the wire or axis that supports the globe will be parallel to the earth's axis, and the north pole of the globe will point toward the north pole of the heaven. The same hour will then be shewn in several of the hollows, by the ends of the shadows of their respective stiles. The axis of the globe will cast a shadow on the same hour of the day, in the equinoctial dial, in the centre of which it is placed, from the 20 th of March to the 22 d of September ; and, if the meridian of your place on the globe be set even with the meridian stile, all the parts of the globe that the sun shines upon, will answer to those places of the real earth which are then enlightened by the sun. The places where the shade is just coming upon the globe, answer all to those places of the earth to which the sun is then setting ; as the places 1 As the declination of the needle is very uncertain, and varies even at the same place, the dial should be rectified by means of a meridian line, drawn upon the side of the window. ED. I 2 DIALING. where it is going off, and the light coming oil, answer to all those places of the earth where the sun is then rising. And, lastly, if the hour of VI be marked on the equator in the meridian of your place (as it is marked on the meridian of London in the figure), the division of the light and shade on the globe will shew the time of the day. The northern stile of the dial (opposite to the southern or meridian one) is hid from sight in the figure, by the axis of the globe. The hours in the hollow to which that stile belongs, are also supposed to be hid by the oblique view of the figure ; but they are the same as the hours in the front hollow. Those also in the right and left hand semicircular hollows are mostly hid from sight ; and so also are all those on the sides next the eye of the four acute angles. The construction of this dial is as follows. See *" Plate XIII. XIII, Sup. ... r . . On a thick square piece or wood, or metal, draw the lines a c and b d, as far from each other as you intend for the thickness of the stile abed, and in the same manner, draw the like thickness of the other three stiles, efg A, ik I m, and nopq, all standing outright as from the centre. With any convenient opening of the com- passes, as a A (so as to leave proper strength of stuff when K I is equal to a A} set one foot in a, as a centre, and with the other foot de- scribe the quadrantal arc Ac. Then, without altering the compasses, set one foot in b as a centre, and with the other foot describe the quadrant d B. All the other quadrants in the figure must be described in the same manner, and with the same opening of the compasses, on their centres e,f; i 9 k ; and n, o : and each DIALING. 133 quadrant divided into six equal parts, for so many hours, as in the figure ; each of which parts must be subdivided into four, for the half hours and quarters. At equal distances from each corner, draw the right lines Jp and Kp, L q and Mq, Nr, and Or, Ps, and Q s ; to form the four angular hollows IpK, LqM, NrO, and PsQ: mak- ing the distances between the tips of the hol- lows, as IK, LM 9 NO, and PQ, each equal to the radius of the quadrants ; and leaving suffi- cient room within the angular points, p, q, r, and s, for the equinoctial circle in the middle. To divide the insides of these angles properly for the hour-spaces thereon, take the following method. Set one foot of the compasses in the point /, as a centre ; and open the other to K, and with that opening describe the arc K t : then, with- out altering the compasses, set one foot in K, and with the other foot describe the arc /f. Di- vide each of these arcs, from / and K to their intersection at t, into four equal parts ; and fro'm their centres 1 and K, through the points of di- vision, draw the right lines 19, 1 4, 1 5, 1 6, 77 ; and K2, Kl, Kl2, Kll ; and they will meet the sides Kp and Ip of the angle Ip AT where the hours thereon must be placed. And these hour-spaces in the arcs must be subdivid- ed into four equal parts, for the half hours and quarters. Do the like for the other three angles, and draw the dotted lines, and set the hours in the insides where those lines meet them, as in the figure : and the like hour-lines will be paral- lel to each other in all the quadrants and in the angles. Mark points for all these hours, on the upper I 3 134 DIALING. side, and cut out all the angular hollows, and the quadrantal ones, quite through the places where their four gnomons must stand ; and lay down the hours on their insides, as in Plate XII, and then set in their four gnomons, which must be as broad as the dial is thick ; and this breadth and thickness must be large enough to keep the shadows of the gnomons from ever falling quite out at the sides of the hollows, even when the sun's declination is at the greatest. Lastly, draw the equinoctial dial in the middle, all the hours of which are equidistant from each other ; and the dial will be finished. As the sun goes round, the broad end of the shadow of the stile abed will shew the hours in the quadrant A c, from sun-rise till VI in the morning ; the shadow from the end M will shew the hours on the side L q from V to IX in the morning ; the shadow of the stile efg h in the quadrant D g (in the long days) will shew the hours from sun-rise till VI in the morning ; and the shadow of the end N will shew the morning hours, on the side r, from III to VII. Just as the shadow of the northern stile abed goes off the quadrant Ac^ the shadow of the southern stile iklm begins to fall within the quadrant F /, at VI in the morning ; and shews the time, in that quadrant, from VI till XII at noon ; and from noon till VI in the evening in the quadrant m E. And the shadow of the end shews the time from XI in the forenoon till III in the afternoon, on the side r N ; as the shadow of the end P shews the time from IX in the morning till I o'clock in the afternoon, on the sides Qs. At noon, when the shadow of the eastern stile efg h goes off the quadrant k C (in which it DIALING. 135 shewed the time from six in the morning till noon, as it did in the quadrant g D from sun- rise till VI in the morning (the shadow of the western stile nopq begins to enter the quadrant Hp ; and shews the hours thereon from XII at noon till VI in the evening ; and after that till sun-set, in the quadrant q G ; and the end Q casts a shadow on the side P s from V in the evening till IX at night, if the sun be not set be- fore that time. The shadow of the end / shews the time on the side K p from III till VII in the afternoon ; and the shadow of the stile abed shews the time from VI in the evening till the sun sets. The shadow of the upright central wire, that supports the globe at top, shews the time of the day, in the middle or equinoctial dial, all the summer half year, when the sun is on the north side of the equator. In this Supplement to my book of Lectures, all the machines that I have added to my appar- atus, since that book was printed, are described, excepting two ; one of which is a model of a mill for sawing timber, and the other is a model of the great engine at London bridge, for raising water. And my reasons for leaving them out are as follows. First, I found it impossible to make such a drawing of the saw-mill as could be understood ; because in whatever view it be taken, a great many parts of it hid others from sight. And, in order to shew it in my Lectures, I am obliged to turn it into all manner of positions. 3 3 For the plan and elevation of a Saw-mill, see Gray's Experienced Mill-wright, lately published, p. 68. For the method 136 DIALING. Secondly, because any person who looks on Fig. 1 of Plate XII in the book, and reads the account of it in the fifth Lecture therein, will be able to form a very good idea of the London- bridge engine, which has only two wheels and two trundles more than there are in Mr. Alder- sea's engine, from vyhich the said figure was taken. method of constructing one, see Wolfii Opera Mathematica, torn, i, p. 694, and Boecklerus's Theatrum Machinarum. An excellent saw-mill was invented by Mr. James Stan- field, in the year 1765, for which he received a reward of one hundred pounds from the society for the encourage- ment of arts. The original mill which Mr. Stanfield con- structed, was worked for five successive years, in conse- quence of successive premiums offered and paid by the so- ciety, amounting, in all, to two hundred and twenty pounds. A description of this machine, illustrated by five folio plates, will be found in Bailey's Designs of Machines, &c. approved and adopted by the society for the encouragement of arts, vol. i, p. 137, ED. fm,.: APPENDIX TO MECHANICS, &c. BY THE EDITOR, APPENDIX TO FERGUSON'S LECTURES MECHANICS. ON THE CONSTRUCTION OF UNDERSHOT WATER WHEELS FOR TURNING MACHINERY. -tA-LTHoucH no country has been more distin- guished than this, by its discoveries and im- provements in the mathematical sciences, yet no- where have these improvements less effectually contributed to the advancement of the mechanic- al arts. The discoveries of our philosophers, particularly in the construction of machinery, have been locked up in the recesses of alge- braical formulae ; and one would have imagined, that they deemed it beneath their dignity to level their speculations to the capacity of common artists. On this account, the mill-wrights of this country are still guided by their own pre- judices ; and, if they are furnished with some useful hints and maxims by the few practical 14O MECHANICS. treaties which are to be met with, they are left in the dark to be directed by their own judg- ment, in the most important parts of the con- struction, la the preceding lectures, Mr. Fer- guson has given some useful directions for the construction of corn mills ; but as these are too limited to be of extensive utility, we shall en- deavour to supply the defect, by treating this important subject at considerable length. Let us begin, then, by shewing the method of construct- ing the mill course, and delivering the water oa the wheel. On the construction of the Mill Course. on the mill As it is of the highest importance to have the height of the fall as great as possible, the bot- tom of the canal, or dam, which conducts the water from the river, should have a very small declivity ; for the height of the water-fall will diminish in proportion as the declivity of the canal is increased. On this account, it will be sufficient to make A B (Fig. l) slope about one inch in 2OO yards, taking care to make the de- clivity about half an inch for the first 48 yards, in order that the water may have a velocity suffi- cient to prevent it from flowing back into the river. The inclination of the fall, represented by the angle OCR, should be 25 5V ; or C R 9 the radius, should be to G R the tangent of this angle, as 100 to 48, or as 25 to 12; and since the surface of the water S b is bent from a b into a c, before it is precipitated down the fall, it will be necessary to incurvate the upper part CD of the course into B Z>, that the water at the bottom may move parallel to the water at 3 MECHANICS. the top of the stream. For this purpose, take the points B, -D, about 12 inches distant from C, and raise the perpendiculars B E^ D E : the point of intersection E will be the centre from which the arch B D is to be described ; the ra- dius being about 10^- inches. Now, in order that the water may act more advantageously up- on the float-boards of the wheel W l^ it must assume a horizontal direction HK^ with the same velocity which it would have acquired when it came to the point G But, in falling from C to G, the water will dash upon the horizontal part jF/G, and thus lose a great part of its velocity ; it will be proper, therefore, to make it move along F H an arch of a circle to which D F and K H are tangents in the points F and H. For this purpose make G F and G H each equal to three feet, and raise the perpendiculars H /, F /, which will intersect one another in the point / distant about 4 feet 9 inches and 4-10 tbs from the points F, and H, and the centre of the arch FH will be determined. The distance H K, through which the water runs before it acts up- on the wheel, should not be less than two or three feet, in order that the different portions of the fluid may have obtained a horizontal direc- tion : and if H K be much larger, the velocity of the stream would be diminished by its fric- tion on the bottom of the course. That no water may escape between the bottom of the course KH and the extremities of the float- boards, K L should be about 3 inches, and the extremity o of the float-board n o should be be- neath the line H K Jf, sufficient room being left between o and M for the play of the wheel, or K. L M may be formed into the arch of a circle KM concentric with the wheel. The line 142 MECHANICS. called by M. Fabre, the course of impulsion (Je coursicr d? impulsion) should be prolonged, so as to support the water as long as it can act up- on the float-boards, and should be about 9 inches distant from P, a horizontal line passing through the lowest point of the fall ; for if L were much less than 9 inches, the water having spent the greater part of its force in impelling the float-boards, would accumulate below the wheel and retard its motion. For the same reason, Another course, which is called by M. Fabre, the course of discharge (le coursier de decharge) should be connected with L M V 9 by the curve F~N, to preserve the remaining velocity of the water, which would otherwise be destroyed by falling perpendicularly from P~to N. The course of discharge is represented by FZ, sloping from the point 0. It should be about 1 d yards long, having an inch of declivity in every two yards. The canal which reconducts the water from the course of discharge to the river, should slope about 4 inches in the first' 200 yards, 3 inches in the second '20O yards, decreasing gradually till it terminates in the river. But if the river to which the water is conveyed, should, when Kwoln by the rains, force the water back upon the wheel, the canal must have a greater decli- vity, in order to prevent this from taking place. Hence it will be evident, that very accurate level- ling is necessary for the proper formation of the mill course. In order to find the breadth of the course of discharge, multiply the quantity of water ex- pended in a second, 1 measured in cubic feet, by - The quantity of water expended in a second may found pretty accurately by measuring the depth of the > be wa- -ter MECHANICS. 143 756, for a first number. Multiply the square root of dK (dK being found by subtracting O K or P R, each equal to a foot, from dO or b R, the height of the fall) by L, or 1. of a foot, and also by 100O, and the product will be a second number. Divide the first number by the second, and the quotient will be nearly the least breadth of the course of discharge. If the breadth of the course, thus found, should be too great or too small, the point L has been placed too far from or too near it. Increase, therefore, or diminish L ; and having sub- tracted from d or b P, the quantity by which K is greater or less than a foot, repeat the operation with this new value of d AT, and a more convenient answer will be foundi The preced- ing rule will give too large a breadth to the course, when the expence of water is great, and the height of the fall inconsiderable. But the course of discharge ought always to have a very considerable breadth, and which should be greater than that of the course of impulsion, that the water having room to spread, may have less depth ; and that a greater height may be procured to the fall, by making OL> and consequently OK, as small as possible ; for the breadth of the course is inversely as O //, that is, it increases as L diminishes, and diminishes as it increases. The reader may tcr .at a, (A B, the bottom of the canal, being nearly horizontal, and its sides perpendicular), and the breadth of the canal at the same place. Take the cube of the depth of the water in feet, and extract the square root of it. Multiply this root by the breadth of the canal, and also by 50/. Divide the product by ]00, and the quo- tient will be the expence of water in a second, measured in cubic feet. This rule is founded on the formula, x=5.O7, b X a* ; where x is the quantity of water expended in a ccond, b the breadth of the canal, and d its depth. .144- MECHANICS. suppose that this rule still leaves us to guess at the breadth of the course of discharge ; but, from the purposes for which it is used, it is easy to know when it is excessively large or small ; and it is only when this is the case, that we have any occasion to seek for another breadth, by tak- ing a new value of L. The section of the fluid at K should be rect- angular, the breadth of the stream having a de- terminate relation to its depth. If there is very much water, the breadth should be triple the depth ; if there is a moderate quantity, the breadth should be double the depth; and, if there is very little water, the breadth and depth should be equal. That this relation may be pre- served, the course at the point K must have a certain breadth, which may be thus found : Divide the square-root of d K (found as before) by the quantity of water expended in a second, and extract the square-root of the quotient. Multiply this root by .623, if the breadth is to be triple the depth ; by .515, if it is to be double; and by .364, if they are to be equal, and the product will be the breadth of the course at K. The depth of the water at ^is therefore known; being either one third, or one half of the breadth of the course, or equal to it, according to the quantity of water furnished by the stream. PLATE T, In Fig. I, b P is called the absolute fall, which is found by levelling. Draw the horizontal lines b d , P ; d will thus be equal to b P, and will likewise be the absolute fall. The relative fall is the distance of the point d from the surface of the water at K, when the depth of the water is considerably less than d K, but is reckoned from the middle of the water at K, when d K is MECHANICS. 145 very small/ The relative fall, therefore, may be determined by subtracting K s which is ge- nerally a foot, from the absolute fall d 0, and by subtracting also either the whole, or one half of he natural depth of the water at K, accord- ing as d K is great or small in proportion to this depth. The next thing to be determined, is the Breadth of breadth of the course at the top of the fall , e th c " se and the breadth of the canal at the same place. O f the fafi. To find this ; multiply the quantity of water ex- pended in a second by 100, for a first number ; take such a quantity a you would wish, for the depth of the water, and, having cubed it, ex- tract its square-root, and multiply this root by 507, for a second number ; divide the first num- ber by the second, and the quotient will be the breadth required. The breadth, thus found, may be too great or too small in relation to the depth. If this be the case, take one half of the breadth, thus found, and add to it the number taken for the depth of the water ; the sum will be the true depth, with which the operation is to be repeat- ed, and the new result will be better proportion- ed than the first. The mill-course being thus constructed, we Quantity of may now find more exactly the quantity of wa- w . a L Cr , f ! ur * 1 r . , i J v- 1. ' nished in a ter furnished in a second, ror this purpose, second, subtract one half the depth of the water at K from d K, and having multiplied the remainder * The depth of the water, here alluded to, is its natural depth, or that which it would have if it did not meet the float-boards. The effective depth is generally two and a half times the natural depth, and is occasioned by the im- pulse of the water on the float-boards, which forces it to swell, and increases its action upon the wheel. Vol. II. K 146 MECHANICS. by .5719, extract the square root of the pro- duct. Multiply this root by the breadth of the course at K multiplied into the depth of the wa- ter there, 3 and the result will be the true ex- pence of the source in cubic feet. In order to know whether the water will have sufficient force to move the least millstone which should be employed, namely, a millstone weigh- ing, along with its axis and trundle, 1550 pounds avoirdupois, take the relative fall increas- ed by one half the natural depth of the water at Fl g-? K 9 viz. dK, and multiply it by the expence of the source in cubic feet ; if the product is 32.95, or above it, the machine will move without in- terruption. If the product be less than this number, the weight of the millstone ought to be less than 1550 pounds, and the meal will not be ground sufficiently fine ; for the resistance of the grain will bear up the millstone, and allow the meal to escape before it is completely ground. As it is of great consequence that none of the water should escape, either below the float-boards, or at their sides, without contributing to turn the wheel, the course of impulsion, KJS, should PLATE i, be wider than the course at /t, as represented in App. Kg. i Fig. 2, where CD, the course of impulsion, cor- *' responds with LV in Fig. 1, AE corresponds with HK, and BC with KL. The breadth of the float-boards, therefore, should be wider than m n, and their extremities should reach a little below 9 like n o in Fig. 1 . When this precaution is taken, no water can escape, without exerting its force upon the float-boards. J That is, by the area of the rectangular section of the gtream at K. MECHANICS. 147 On the size of the wafer wheel, and on the number, magnitude, and position, of its float-boards. The diameter of the wheel should be as great size of the as possible, unless some particular circumstances w * tcr , * . t . waeel. in the construction prevent it ; but ought never to be less than seven times the natural depth of the stream at K, the bottom of the course. 4 It has been much disputed among philosophers, whe- ther the wheel should be furnished with a small or a great number of float-boards. M. Pitot has shewn, that when the float-boards havp different degrees of obliquity, the force of impulsion upon the different surfaces will be reciprocally as their breadth : thus, in Fig. 3, the force upon k e will PLATE i. be to the force upon DO as DO to h e. s He there- fore concludes, that the distance between the Number of float-boards should be equal to one half of the float * l * >ari * ..... . . . according arch plunged in the stream, or that, when one is to at the bottom of the wheel, and perpendicular to the current, as DE, the preceding float-board BC should be leaving the stream, and the suc- ceeding one FG just entering into it. 6 For, when the three float-boards FG, DE, BC, have the same position as in the figure, the whole force of the current JVM will act upon DE, having the most advantageous position for receiving it : 4 The diameter here meant is double the mean radius, or the distance between the centre of the wheel and the middle of the natural stream, which impels it, or what is called the centre of impulsion. By adding or subtracting the half of the stream's natural depth, to or from the mean radius, we have the exterior and interior radius of the wheel. J See Traite d'Hydrodynamique, $771. 6 Mem. de 1'Acad. Paris. 1729, 8", p. 3 59. K 2 148 MECHANICS. whereas, if another float-board d e were inserted between FG and DE, the part i g would cover jDO, and, by thus substituting an oblique for a perpendicular surface, the effect would be dimi- nished in the proportion of DO to i g. Upon this principle it is evident, that the depth of the float-board DE should always be equal to the versed sine of the arch between any two float- boards, DE being the versed sine of EG. For the use of those who may wish to follow M. Pitot, though we are of opinion that he recommends too small a number of floats, we have calculated the following table upon the above principles. It exhibits the proper diameter of water wheels, the number of float-boards they should con- tain, and the size of the float-boards, when any two of these quantities are given. According to M. Pitot, the proper relation between these is of so great importance, that if a water wheel, 16 feet diameter, with its float-boards three feet deep, should have nine instead of seven, one twelfth of the whole force of impulsion would be lost. 7 7 Desaguliers has adopted the rule given by Pitot. Sec his Experimental Philosophy, vol. ii, p. 424. 149 Table of the number of 'float-boards in undershot wheels. Diam r of the wheel in feet. Depth of the float-boards in feet. 1 1.5 2 2.5 3 3.5 4 10 10 8 7 6 5 5 5 11 10 8 7 6 5 5 5 12 11 9 8 7 6 6 5 13 11 9 8 7 6 6 5 14 12 9 8 7 7 6 6 15 12 9 8 7 7 6 6 16 12 10 9 8 7 7 6 17 12 10 9 8 7 7 6 18 13 11 9 8 8 7 6 19 13 11 10 9 8 7 7 20 14 11 1O 9 8 7 7 21 14 12 10 9 8 7 7 22 15 12 10 9 8 8 7 23 15 12 10 9 8 8 V 24 15 12 11 10 9 8 8 25 16 IS 11 10 9 8 8 26 16 13 11 10 9 8 8 27 16 13 11 10 9 8 8 28 17 13 12 10 9 9 8 29 17 14 12 11 10 9 8 30 17 14 12 11 1O 9 9 32 18 14 12 11 10 9 9 K 3 ISO MECHANICS. In order to find from the preceding table the number of float-boards for a wheel 20 feet in dia- meter, (the diameter of the wheel being reckon- ed from the extremity of the float-boards), their depth being two feet; enter the left hand column with the number 2O, and the top of the table with the number 2, and in a line with these num- bers will be found 1O, the number of float-boards which such a wheel would require. As the numbers representing the depths of the float-boards, and the diameter of the wheel, increase more rapidly than the numbers in the other columns, the preceding table will not shew us with accuracy the diameter of the wheel when the number and depth of the float-boards are given ; ten float-boards, for example, two feet deep, answering to awheel either 19, 2O, 21, 22, or 23 feet diameter. This defect, however, may be supplied by the following method. Di- vide 360 degrees by the number of float-boards, and the quotient will be the arch between each. Find the natural versed sine of this arch, and say, as 1OOO is to this versed sine, so is the wheeFs radius to the depth of the float-boards ; and to find the diameter of the wheel, say, as the above versed sine is to 10OO, so is the depth of the float-boards to the wheel's radius. The rule We have already said, that the number of ^^'""floatboards found by the preceding table is too small. Let us attend to this point, as it is of con- siderable importance. It is evident from Fig. 3, that when one of the floats, as DE, is perpendi- Pt'TBi. cular to the stream, it receives the whole im- F>s ' 3 ' pulse of the water in the most advantageous manner ; but when it arrives at the position d v. 51. ' Traite d'Hydrodynramique, notes on chap, x j also 778. 152 MECHANICS. number of floats is determined. Having fixed upon the radius and velocity of the wheel, and on the portion of its circumference that ought to be plunged in the stream, he imagines the wheel to have different numbers of float-boards, and then computes the momentum of the water a- gainst all the parts of those that are immersed. The number of float-boards which gives the greatest momentum should be adopted as the most advantageous. When the velocity of the stream was thrice that of the wheel,, and when 72 degrees of the circumference were immersed, Bossut found that the number of float-boards should be 36. When a greater arch is plunged in the stream, the velocity continuing the same, the number should be increased, and vice versa. The fleaty This rule, however, is too difficult to be of use b u oar ?j u to the practical mechanic. From what has been should be . . . r . . , , . , as numer- said, it is evident, that in order to remove any cus as pos- inequality of motion in the wheel, and prevent sible. J r . i * , . r - . the water from escaping beneath the tips of the float-boards, the wheel should be furnished with the greatest number of float-boards possible, with- out loading it, or weakening the rim on which they are placed. 1 This rule was first given by Dupetit Vandin,* and afterwards by M. Fabre, 5 and it is not difficult to see, that if the millwright should err in furnishing the wheel with too many float-boards, the error will be perfectly trifling, and that he wouM lose much more by erring on the other side. The float-boards should not be rectan- 1 Brisson (Traite Elementaire de Physique) observes, that there should be 48 floats, instead of 40, as generally used in a wheel 20 feet in diameter. 1 Mem. des Savans Etrangers, torn. i. 3 Sur les Machines Hydrauliques, p. 55, N. 1Q3 MECHANICS. 153 gular, like a b n c in Fig. 3, but should be bevelled Fg- 3- like a b m c. For if they were rectangular, the ex- tremity b n would interrupt a portion of the water, which would otherwise fall on the corresponding part of the preceding float-board. The angle a b m may be found thus. Subtract from 18O the number of degrees contained in the immersed arch CEG, and the half of the remainder will be the angle required. It has been already observ- ed, that the effective depth of the water at K (Fig. 1) is generally two and a half times greater than the natural depth. The height Z), there- fore, of the float boards should be two and a half rimes the natural depth of the current at K. % >. The breadth of the float-boards should always be a little greater than the breadth of the course at K 9 the method of finding which has been already pointed out. M. Pitot has shewn, 4 that the float-boards inclination should be perpendicular to the rim, or, in other ^^ words, a continuation of the radius. This, in- boards, deed, is true in theory, but it appears from the most unquestionable experiments, that they should be inclined to the radius. This was dis- covered by Deparcieux, in 1753, (not in 1759, as Fabre asserts), who shews, that the water will thus heap up on the float-boards, and act, not only by its impulse, but also by its weight. 5 This discovery has been confirmed also by the Abbe Bossut, 6 who found, that when the velo- city of the water is about 3 ^ of a foot, or 1 1 feet per second, the inclination of the float-board to 4 Mem. Acad. Royale 1729, 8", p. 350. * Mem. de 1'Acad. 1754, 4'", p. 614, 8", p. 944. r Traite d'Hydrodynamique, ^ 814 and ^ 817. 154 MECHANICS. the radius should be between 1 5 and 30 degrees. M. Fabre, however, is of opinion, that when the velocity of the stream is 1 1 feet per second, or above this, the inclination should never be less than 30 degrees ; that when this velocity dimi- nishes, the inclination should diminish in propor- tion ; and that when it is four feet, or under, the inclination should be nothing. -In order to find the inclination for wheels of different radii, let Kg.?. AH (Fig. 3) be the radius, bisect. P H, the height of the float-board, in z, and having drawn P K perpendicular to PA, set off P K equal to Pi, and join UK; HK will be the position of the float-board inclined to the radius AH by the angle KHP. This construction supposes the greatest value of the angle KHP to be 26' 34'. On the formation of the spur wheel and trundle. size of the The radius of the spur wheel is found by s P urwhed - multiplying the mean radius of the water-wheel by that of the lantern, which may be of any size, and also by the number of turns, which the spindle or axis of the lantern performs in a se- cond, 7 and then by the number 2.151. This product being divided by the square-root of the relative fall, the quotient will be the r?dius re- quired. The number of teeth in the wheel should be to the number of staves in the trundle Number of as their respective radii. In order to find the teeth in the wheel nf\d * trundle. 7 The method of determining the velocity of the spin- dle, or the mill-stone, will be afterwards pointed out. The axis of the lantern should, in general, make about $O turn& in a minute. MECHANICS. 155 exact number, take the proper diameter of the teeth and the staves, which ought to be two and a half inches each in common machines, and de- termine ako how much is to be allowed for the play of the teeth, which should be about two and a half tenths of an inch ; add these three numbers, and divide by this sum the mean cir- cumference of the spur wheel, 8 the quotient will be nearly the number of teeth in the wheel. Let us call this quotient x^ to avoid circumlocu- tion. Multiply x by the mean radius of the trundle, and divide the product by the radius of the spur wheel. If the quotient is a whole num- ber, it will be the exact number of staves in the trundle, and #, if it were an integer, will be the exact number of teeth in the wheel. But should the quotient be a 'mixed number, diminish the integer, which may still be called JT, by the num- bers 1, 2, 3, &c. successively, and at every di- minution, multiply ,r, thus diminished, by the radius of the trundle, and divide the product by the radius of the wheel. If any of these opera- tions give a quotient without a remainder, this quotient will be the number of staves in the trundle, and .r, diminished by one or more units, will be the number of teeth in the wheel. Thus let the radius of the trundle be one foot, that of the wheel four feet, the thickness of the teeth and the staves two and a half inches, or T 3 ^ of a foot, and the space for the play of the teeth two and a half tenths of an inch, or -^ ; the suni of the three quantities will be -^-^ or -^ of a foot * f and '25 feet, or u~. o f a foot, the circumference The mean radius is reckoned from the centre of the rheel to the centre of the teeth. 156 MECHANICS. of the wheel, divided by -^ will give a ^ a , or 57-^|- feet. Multiply the integer a? or 57 by 1, the radius of the lantern ; but as the product 57 will not divide by 4, the radius of the wheel, let us diminish x, or 57, by unity, and the re- mainder 56 being multiplied by 1 , the radius of the trundle, and divided by four, the radius of the wheel, gives 14 without a remainder, which, therefore, will be the number of staves, while 56, or x diminished by unity, is the number of teeth in the spur-wheel. Had it been possible to make the number of teeth equal to 57-|-f-, 2 j- inches would be the pro- per thickness for the teeth and the staves ; but, as the number mus be diminished to 56, there will be an interval left, which must be distribut- ed among the teeth and staves, so that a small addition must be made to each. To do this, divide the circumference of the wheel ^-~ of a foot by the number of teeth 56, and, from the quotient r * 5 subtract the interval for the play of the teeth -\-$ or I * , the remainder ~~ being halved, will give ~ ^ of a foot, or 2 inches and 5.8 tenths, for the thickness of every tooth and stave, ~^ of an inch being added to each tooth and stave to fill up the interval. It may sometimes happen, however, that, in diminishing x successively by unity, a quotient will never be found without a remainder. When this is the case, seek out the mixed number which approaches nearest an integer, and take the integer to which it approximates for the number of staves in the lantern. Thus, when the radius of the wheel is 4- feet, the different quotients obtained, after diminishing x by one, two,three, four, will be 1- and 13- MECHANICS. 157 an integer is 13~?-> being only y^ less than 14, which will therefore be the number of staves in the trundle. 9 ^ In a succeeding article on the teeth of wheels, Form of the we have shewn what form must be given them teetlu in order to produce an uniformity of action. The following method, however, will be pretty accurate for common works. In Plate IV, PLATS IY, Fig. 7, take E, equal to the radius of the Flg ' 7 ' trundle, 1 and desdribe the acting part BA^ and with the same radius describe CD. When the teeth of the wheel are perpendicular to its plane, as in the spur wheels of corn mills, we must Bi- sect CD in n, and drawing mn perpendicular to ED, make the plane BACD move round upon mn as an axis ; the figure thus generated like abc^ Fig. 8, will be the proper shape for FJ g-8. the teeth. The pivots-, or gudgeons, on which vertical size of the L i j i i it i i gudgeons. axes move, should be conical ; and those which are attached to horizontal arbors, should be cy- lindrical, and as small and short as possible. A gudgeon two inches in diameter will support a weight of 3139 pounds avoirdupois, though we often meet with gudgeons three or four inches in diameter, when the weight to be supported is considerably less. By attending to this, the fric- tion of the gudgeons will be much diminished, and the machine greatly improved. Particular care, too, should be taken, that the axis of the gudgeons be exactly in a line with the axis of 9 See Fabre sur les Machines Hydrauliques, p. 30-1, 546. 1 The staves of tha trundle should be as short a* possible. i 158 MECHANICS* the arbor which they support,* otherwise the action or motion of the wheels which they carry will be affected with periodical inequalities, On the formation^ size, and velocity , of the mill- stone, &c. tin the nv In the fourth lecture, 5 Mr. Ferguson has given e several useful directions for the formation of the grinding surfaces of the millstones ; to which we have only to add, that when the furrows are worn shallow, and consequently new dressed with the chisel, the same quantity of stone must be taken from every part of the grinding sur- face, that it may have the same convexity or concavity as before. As the upper millstone should always have the same weight when its velocity remains unchanged, it will be necessary to add to it as much weight as it lost in the dressing. This will be most conveniently done by covering its top with a layer of plaster, of the same diameter as the layer of stone taken from its grinding surface, and as much thicker than the layer of stone, as the specific gravity of the stone exceeds the specific gravity of the plaster.* That the reader may have some idea of the man- ner in which the furrows, or channels, are ar- i, ranged, we have represented in Plate 1 , Fig. 4, ri - 4- the grinding surface of the upper millstone,, upon 1 The diameter of the gudgeon must be proportional to the square -root of the weight which it supports. J Vol. i, p. 85. 4 The relative weights of the stone and plaster may be determined from the table of specific gravities at the end of vol. i. MECHANICS. 1,59 the supposition that it moves from east to west, or for what is called a right-handed mill. When the millstone moves in the opposite direction, the position of the furrows must be reversed. In Fig. 5 we have a section of the millstone Fg- 5- spindle and lantern. The under millstone MPHG, which never moves, may be of any thickness. Its grinding surface must be of a conical form, the point b being about an inch above the horizontal line PR, and Ma and Pb being straight lines. The upper millstone EFPM, which is fixed to the spindle CD at C, and is carried round with it, should be so hol- lowed that the angle OMa, formed by the grinding surfaces, may be of such a size that n being taken equal to n M, n s may be equal to the thickness of a grain of corn. 5 The dia- meter N of the mill eye mC should be be- tween 8 and 14 inches ; and the weight of the upper millstone E P joined to the weight of the spindle CD and the trundle x (the sum of which three numbers is called the equipage of the turn- ing millstone), should never be less than 155O pounds avoirdupois, otherwise the resistance of the grain would bear up the millstone, and the meal be ground too coarse. In order to find the weight of the equipage ; vv c i g i lt ot Divide the third of the radius of the gudgeon by the radius of the water-wheel which it sup- ports, and having taken the quotient from 2.25, * Jn note 6, p. 93, vol. i, we have said that the core does not begin to be ground till it has insinuated itself as far as two thirds of the radius, or the centre of gyration .; but for reasons which may be seen in Fabre sur let Ma- chines Hydrauliques, p. 238, the grinding should commence at the point n, equidistant from and M, 16O MECHANICS. multiply the remainder by the expence of the source, by the relative fall, and by the number 19911, and you will have a first quantity, which may be regarded as pounds. Multiply the square root of the relative fall by the weight of the arbor of the water wheel, by the radius o its gudgeon, and by the number 1617, and a se- cond quantity will be had, which will also re- present pounds. Divide the third part of the radius of the gudgeon by the radius of the water- wheel, and having augmented the quotient by unity, multiply the sum by 1005, and a third quantity will be obtained. Subtract the second quantity from the first, divide the remainder by the third, and the quotient will express the num- ber of pounds in the equipage of the millstone. The weight of the equipage being thus found, extract its square root, expressed in pounds, and multiply it by .039, and the product will be the radius of the millstone in feet/ size of the i n order to find the weight and thickness of the upper millstone, the following rules must be observed. 1. To find the weight of a quantity of stone equal to the mill eye ; Take any quantity which seems most pfoper for the weight of the spindle CD and the lantern JT, and subtract this quan- tity from the weight of the millstone's equipage, for a first quantity. Find the area of the mill eye, and multiply it by the weight of a cubic foot of stone of the same kind as the millstone, (found from the table of specific gravities, vol. i) and a second quantity will be had. Multiply 6 This rule supposes, that when the diameter of the millstone is 5 feet, the weight of the equipage should be 4307 avoirdupois pounds. MECHANICS. 161 the area of the millstone by the weight of a cu- bic foot of the same stone, for a third quantity. Multiply the first quantity by the second, and divide the product by the third, and the quotient will be the weight required. 2. To find the number of cubic feet in the turning millstone, supposing it to have no eye ; From the weight of the spindle and lantern subtract the quantity found by the preceding rule, for the first number. Subtract this first number from the weight of the equipage, and a second number will be obtained. Divide this second quantity by the weight of a cubic foot of stone of the same quality as the millstone, and the quotient will be the number of cubic feet Fig. 5. in E MPF, m C being supposed to be filled up. 3. To find the quantities m N and E M, i. e. the thickness of the millstone at its centre and circumference ; Divide the solid content of the millstone, as found by the preceding rule, by its area, and you will have a first quantity. Add b R, which is generally about an inch, to twice the diameter of a grain of corn, for a second quantity. Add the first quantity to one third of the second, and the sum will be the thickness of the millstone at the circumference. Subtract the third of the second quantity from the first quantity, and the remainder will be its thickness at the centre. 7 The size of the mill-stone being thus found, Velocity of its velocity is next to be determined. M. Fab observes, that the flour is the best possible when a millstone 5 feet in diameter makes from 48 to 61 revolutions in a second. Mr. Ferguson al- * These rules are founded upon formulae, which may be icen in Fabre tur les Machines Hydrauliquct, pp. 172, 2i*9. Vel II. L 163 MECHANICS. lows 60 turns to a millstone 6 feet in diame- ter, and Mr. Imison 12O to a millstone 4-^ feet in diameter. In mills upon Mr. Imison's con- struction, the great heat that must be generated by such a rapid motion of the millstone, must render the meal of a very inferior quality : much time, on the contrary, will be lost, when such a slow motion is employed as is recommended by M. Fabre and Mr. Ferguson. In the best corn mills in this country, a millstone 5 feet in diameter revolves, at an average, 90 times in a minute. 5 The number of revolutions in a se- cond, therefore, which must be assigned to mill- stones of a different size, may be found by di- viding 4pO by the diameter of the millstone in feet. Spindle. The spindle cD, which is commonly 6 feet long, may be made either of iron or wood. When it is of iron, and the weight of the mill- stone 7558 pounds avoirdupois, it is generally three inches in diameter ; and when made of wood, it is 10 or 11 inches in diameter. For millstones of a different weight, the thickness of the spindle may be found by proportioning it to the square root of the millstone's weight, or, which is nearly the same thing, to the weight of the millstone's equipage. pivots, The greatest diameter of the pivot D, upon which the millstone rests, should be propor- tional to the square root of the equipage, a pivot half an inch diameter being able to support an equipage of 5398 pounds. In most machines, 5 Mr. Fenwick of Newcastle, an excellent practical mechanic, observes, that, in the best corn mills in Eng- land, mill-stones from 4% to 5 feet in diameter revolve from 90 to 100 times in a second. .MECHANICS. the diameter of the pivots is by far too large, being capable of supporting a much greater weight than they are obliged to bear. The fric- tion is therefore increased, and the performance of the machine diminished. The bridge-tree AB is generally from eight Fig. g . to 1O feet long, and should always be elastic, " e dge " that it may yield to the oscillatory motion of the mill-stone. 6 When its length is 9 feet, and the weight of the equipage 5 1 82 pounds, it should be 6 inches square ; and when the length remains unchanged, and the equipage varies, the thickness of the bridge-tree should be propor- tional to the square root of the equipage. On tlie performance of Undershot Mills. The performance of any machine may be Perform- properly represented by the number of pounds which it will elevate, in a given time, by means mills, of a rope KL (Fig. 5), wound upon the spindle CD, and passing over the pulley L. 7 In order to find the weight which a given machine will raise ; Divide the third part of the radius of the gudgeon of the water-wheel, by the mean radius of the wheel itself, and, having subtracted the quotient from 2.25, multiply the remainder by the expence of water in a second in cubic feet, by the height of the relative fail, and by the number 19911, for a first quantity. Multiply the weight of the arbor of the water-wheel, and 6 See SeKJor, Architecture Hydraulique, 638 ; or Desa- guliers* Exper. Philos. vol. ii, p. 429. 7 It was in this way that Smeaton measured the per- formance of his models. L 2 164 MECHANICS.. its appendages (viz. the water-wheel itself and the spur wheel), by the radius of the gudgeon in decimals of a foot, by the square root of the relative fall, and the number 1617, and divide the product by the mean radius of the water wheel, and a second quantity will be had. Di- vide the third part of the gudgeon's radius by the mean radius of the water-wheel, augment the quotient by unity, and multiply the sum by the radius of the spindle CD for a third quan- tity. Subtract the second quantity from the first, and divide the remainder by the third quantity, the quotient will be the number of pounds which the machine will raise* Multiply the diameter of the spindle CD by 3.1416, and you will have a quantity equal to the height which W will rise by one turn of the spindle;, this quantity, therefore, being multiplied by the num- ber of turns which the spindle performs in a mi- nute, will give the height through which the weight ^Fwill rise in the space of a minute. According Mr. Fenwick 8 found, by a variety of accurate Fcn " ex P er i m ents made upon good corn-mills, whose upper millstone, being from 4f to 5 feet in dia- meter, revolved from QO to 100 times in a mi- nute, that a mill, or any power capable of raising 30O pounds avoirdupois with a velocity of 21O feet per minute, will grind one boll, of good corn in an hour ; and that two, three, four, or five bolls will be ground in an hour, when a weight of 300 pounds is raised with a velocity of 350 506, 677, or 865 feet respectively in a minute. 9 8 Four Essays on Practical Mechanics, 2* edit. 1802, p. 60. 9 As the differences of these numbers increase nearly by 16, they may be continued by alwajs augmenting the difference MECHANICS* 165 Or, to arrange the numbers more properly : Number of bolls ground in an 2 350 3 506 4 677 5 865 6 1069 Number of feet through which SOOlb. is raised in a minute . . . 210 Supposing it, therefore, to be found, by the preceding rules, that a mill would raise 600 pounds through 253 feet in a minute of time, we have 300 : 600zr253 : 506 ; that is, the same power that can raise 600 pounds through 253 feet, will raise 30O pounds through 506 feet, consequently such a mill will be able to grind three bolls of corn in an hour. 1 According to M. Fabre, the quantity of meal According ground in an hour may be determined by mul- toM re * tiplying 62.4 Paris pounds by the square of the radius of the millstone, and the product will be the number of pounds of meal. But, as this rule is founded upon an erroneous supposition, that the quality of the flour is best when a mill- stone, 5 feet in diameter, performs 48 revolu- tions in a minute, we have made the calculation difference between the two last numbers by i6, and add ing the difference thus augmented to the last number, for the number required. Thus, by adding 16 to 188, the difference between 6/7 and 8G'S, we have 204, which being added to 865, gives 1069 for the number of feet, nearly, through which the power must be able to raise a weight of 30O pounds in a minute, in order to grind six bolls of corn in an hour. 1 The proper result of Mr. Fenwick's experiment was, and made it 21O feet pej: minute. to the edi- tor. 166 MECHANICS. anew, upon the supposition, that the velocity of a millstone, five feet diameter, should be 90 According revolutions in a minute, and have found, that, when mills are constructed upon this principle, the quantity of flour ground in an hour, in pounds avoirdupois, will be equal to the product of the square of the millstone's radius^ and the number 125. <. ILH! 1 3 . J"i ; ; ..: % O~T<} 'The following important maxims have been deduced from 'Mr. Smeaton's accurate experi- ments on undershot mills, and merit the atten- tion of every practical mechanic. Maxim \. That the virtual or effective head of water being the same,* the effect will be near- Maxims, ly as the quantity of water expended. That is, if a mill, driven by a fall of water whose virtual head is 10 feet, and which discharges 30 cubic feet of water in a second, grinds four bolls in an hour ; another mill having the same virtual head, but which discharges 60 cubic feet of wa- ter, will grind eight bolls of corn in an hour. Maxim 2. That the expence of water being the same, the effect will be nearly as the height of the virtual or effective head. 1 The virtual, or effective head of water moving with a certain velocity, is equal to the height from which a heavy body must fall in order to acquire the same velocity. The height of the virtual head, therefore, may be easily deter- mined from the water's velocity, for the heights are as the squares of the velocities, and the velocities, consequently, as the square roots of the heights. Mr. Smeaton ob- serves, that, in the large openings of mills and sluices* where great quantities of water are discharged from mode- rate heads, the real'head of water, and the virtual head, as determined from the velocity, will nearly agree. See his Experiments on Mills, p. 23. r-* MBCHANICS. 167 Maxim 3. That the quantity of water expend- ed being the same, the effect is nearly as the square of its velocity. That is, if a mill, driven by a certain quantity of water, moving with the velocity of four feet per second, grinds three bolls of corn in an hour ; another mill, driven by the same quantity of water, moving with the velocity of five feet per second, will grind nearly 4^ bolls of corn in an hour, because 3 : 4^-^:4 * : 5* nearly, that is, as 16 to 25, the squares of the respective velocities of the water. Maxim 4, The aperture being the same, the effect will be nearly as the cube of the velocity of the water. That is, if a mill driven by water, moving through a certain aperture, with the velocity of four feet per second, grind three bolls of corn in an hour; another mill driven with water, moving through the same aperture with the velocity of five feet per second, will grind 5-f-|- bolls nearly in an hour, for 3 : 5^ 4 3 :5 3 nearly, that is as 64 to 125, the cubes of the water's respective velocities. On the method of constructing Mill-wrights* r Tables, on new principles. Although a mill-wright's table has been con- Construe- structed by Mr. Ferguson, 5 and afterwards alter- tlon of .,1 , ,..-;-__ o. new radl- ed a little by Mr. Imison, so tar as concerns the ^rights' velocity of the millstone ; yet* as we shall now table * shew, the principles upon which it is computed are far from being correct. It is evident that the 5 See vol. j, p. 97, 168 MECHANICS. great wheel must always move with less velocity than the water, even when there is no work to be performed; for a part of the impelling power is ne- cessarily spent in overcoming the inertia of the wheel itself; and if the wheel has little or no velo- city, it is equally manifest that it will produce a , . very small effect. There is consequently a certain Relative ' . . i r i velocity of proportion between the velocity or the water and th d th ter t ^ le W ^ ee ^> wnen tne effect is a maximum* Parent wheel. 6 and Pitot found this proportion to be as 1 to 3 j and Desaguliers, 6 Maclaurin, * and Ferguson, have adopted their determination. 8 But Mr. Smeaton has shewn, that instead of the wheel moving with ^ of the velocity of the water, when the effect is a maximum, as Parent imagined, the greatest effect is produced when the velocity of the wheel is between ~ and Y> the maximum. being much nearer to than f. He observes also, that y would be the true maximum ' if no- ' thing were lost by the resistance of the air, the ' scattering of the water carried up by the wheel, ' and thrown off by the centrifugal force, &c. ' all which tend to diminish the effect more at ' what would be the maximum if these did not 6 take place, than they do when the motion is a * little slower.' 9 But in making this alteration 6 Desaguliers' Experimental Philosophy, vol. ii, p. 424, Lect. 12. 7 Maclaurin's Fluxions, Art. QO?, p. 728. 8 M. Lambert has also adopted the determination of Parent, in his Memoir on Undershot Mills in the Nouv. Mem. de fAcad. de Berlin, 1775, p. 63. 9 Smeaton on Mills, p. 77. M. Bossut and M. Fabre, along with Smeaton, make the velocity of the wheel f ot" the velocity of the water. See Traite d'Hydrodynamique par Bossut, 808.9> Fabre, 66. The great hydraulic MECHANICS. 169 we are warranted not merely by the results of Mr. Smeaton's experiments, but also by deductions of theory. In the investigations from which Parent and Pitot concluded that the velocity of the wheel should be j of the velocity of the water in order to produce a maxi- mum effect, they considered the impulse of the stream upon one float-board only, and therefore made the force of impulsion proportional to the square of the difference between the velocities of m the stream and the float-board. The action of the current, however, is not confined to one float-board, but is exerted on several at the same time, so that the float-boards which are accurate- ly fitted to the mill-course, abstract from the water its excess of velocity, and the force of im- pulsion becomes proportional only to the differ- ence between the velocities of the stream and the float-boards. From this circumstance, the Chevalier de Borda has shewn in his Memoire. sar les Roues Hydrauliquesy that in theory the machine at Marly was found to produce a maximum effect when the velocity of the wheel was that of the cur- rent. 1 Memoires de 1'Acad. Paris, 1/6/, 4to, p. 285. Although the memoir of the Chevalier de Borda was published so early as l/67> yet, in the year 1793, there appeared in the Transactions of the American Philosophi- cal Society, vol. Hi, p. 144, a paper, by Mr. Waring, con- taining the same observations on the maximum effect of undershot wheels. We would willingly believe that Mr. Waring was guided solely by his own investigations ; and that the similarity between his memoir and that of Borda's, was owing to a casual coincidence of sentiment. Unfor- tunately, however, in the same volume of the American Transactions, Mr. Waring describes an improvement, by a Mr. Rumsey, on Barker's mill, which was published in t/75, by M. Mathon clc'la Cour in Roziei's Journal dc 17O MECHANICS. velocity of the wheel is ~ that of the current, and that in practice it is never more than |- of the stream's velocity, when the effect is a maxi- mum. The constant number, too, which is used by Mr. Ferguson for finding the velocity of the water from the height of the fall, viz. 64.2882 is not correct. From the recent experiments of Mr. Whitehurst on pendulums, it appears, that a heavy body falls 16.087 feet in a second of time ; consequently the constant number should be 64.348. In Mr. Ferguson's table, the velocity of the millstone is too small ; and Mr. Imison, in cor- recting this mistake, has made the velocity too great. From this circumstance, the mill-wrights* table, as hitherto published, is fundamentally er- roneous, and is more calculated to mislead than to direct the practical mechanic. Proceeding, therefore, upon the practical deductions of Smea- ton, as confirmed by theory, and employing a more correct constant number, and a more suitable velocity for the millstone, we may construct a new mill-wrights' table by the following rules. Method of 1 5 Find the perpendicular height of the fall of construct- water in feet above the bottom of the mill- C urse at K ( Fi g- 1 9 plate 1 )> and having di- Fig. i. minished this number by one half of the natural depth of the water at K, call that the height of the fall.* Physique. Such unequivocal instances of literary robbery cannot be too severely reprobated. 1 The height of the fall here meant is the relative or virtual height, and it is supposed that the mill-course is so accurately constructed, that the water will have the same velocity at K as it would have at R by falling perpendicu* MECHANICS. 2, Since bodies acquire a velocity of 32.174? feet in a second, by falling through 16.087 feet, and since the velocities of falling bodies are as the square roots of the heights through -which they fall, the square root of 1 6.087 will be to the square root of the height of the fall as 32.174 to a fourth number, which will be the velocity of the water. Therefore the velocity of the water may be always found by multiplying 32.174, by the square root of the height of the fall, and dividing that product by the square root of 1 6.087. Or it may be found more easily by multiplying the height of the fall by the constant number 64.343, and extracting the square root of the product, which, abstracting the effects of friction, will be the velocity of the water requir- ed. 3 larly through CR. This will be nearly the case when the mill-course is 'formed according to the directions formerly given ; though in general a few inches should be taken from the fall, in order to obtain accurately its relative or virtual height. 3 That the velocity of the water is equal to the square root of the product of the height of the fall, and the constant number 64.348, may be shewn in the following manner. Let x be the -velocity of the water, m the height of the fall, = 16.087, and consequently 2 a = 32. 1/4. Then by the first part of the second rule ^/a : y^=r2 a : x therefore * r=- ~ j multiplying by \fa we have x */ a = 2 a A/ m ; putting all the quantities under the radical sign there comes out ^ x 1 a = 4 a 1 m\ extracting the square root of both sides, we have x 1 a 4 a* m, divid- ing by a gives x* = 4 a m or x =r \/ 4 a m. But since the constant number 64.348 is double of 32.174, it will b* equal to 4 o ; then by the latter part of rule second we have x = / 4 a m, which is the same value of x, as wa$ found from the first part of the rule. 172 MECHANICS* 3, Take one half of the velocity of the water, and it will be the velocity which must be giveii to the float-boards, or the number of feet they must move through in a second, in order that the greatest effect may be produced. 4, Divide the circumference of the wheel by the velocity of its float-boards per second, and the quotient will be the number of seconds in which the wheel revolves. 5, Divide 60 by this last number, and the quotient will be the number of revolutions which the wheel performs in a minute. Or the num- ber of revolutions performed by the wheel in a minute, may be found by multiplying the velo- city of the float-boards by 60, and dividing the product by the circumference of the wheel, which in the present case is 47.12. 6, Divide 90 (the number of revolutions which a millstone 5 feet diameter should per- form in a minute) by the number of revolu- tions made by the wheel in a minute, and the quotient will be the number of turns which the millstone ought to make for one revolution of the wheel. 7, Then, as the number of revolutions of the wheel in a minute is to the number of revolu- tions of the millstone in a minute, so must the number of staves in the trundle be to the num- ber of teeth in the wheel, in the nearest whole numbers that can be found. 4 4 We have filled up the sixth column of the tables in the common way ; but, for the proper method of finding the relation between the radius of the spur wheel and trundle, and the exact number of teeth in the one, and staves in the other, we must refer the reader to pp. 154-5 of this volume. MECHANICS. 173 8, Multiply the number of revolutions per- formed by the wheel in a minute, by the num- ber of revolutions made by the millstone for one of the wheel, and the product will be the number of revolutions performed by the millstone in a minute. In this manner the following table has been calculated for a water-wheel fifteen feet in dia- meter, which is a good medium size, the mill- stone being five feet in diameter, and revolving 90 times in a minute. MECHANICS. TABLE I. A NEW MILL-fVRIGHT'S TABLE, In which the velocity of the wheel is one-half t^e velocity oftkt stream, the effects of friction not being considered. \ J Velocity ; Velocity Revolu- of the of the tions of Revolu- tions of Teeth in the wheel Revolu- tions of w 1 11 11 H? a > water per wheel per he wheel second, second, per friction being minute, not one half its dia- being ( that of meter consider- 1 the being 15 ed. water. feet. the mill- stone for one of the wheel. and staves in the trundle. the mill- stone per minute by these staves and teeth. u *J 4-J -< O rt o . P<<~ tJ O 2-Sj *$ si . Mi-d S-B < > S-s (2 S's ~ S ^ "3 8,8 > Q ft ^ ~ ' o rJ H Revol. TOO parts of a revol. 1 8.02 4.01 5.1O 17.65 106 6 90.O1 2 11.34 5.67 7.22 12.47 87 7 90.O3 3 13.89 6.95 8.85 10.17 81 8 90.00 4 16.04 8.02 10.20 8.82 79 9 89.96 5 17.94 8.97 11.43 7.87 71 9 89.95 6 19.65 9.82 12.50 7.20 65 9 90.0O 7 21.22 10.61 13.51 6.66 60 9 89.Q8 8 22.69 11.34 14.45 6.23 56 9 90.02 9 24.06 12.03 15.31 5.88 53 9 9O.02 10 25.37 12.69 16.17 5.57 56 10 90.O6 11 26.60 13.30 16.95 5.31 53 10 90.00 12 27.79 13.90 17-70 5.08 51 10 89.Q1 13 28.92 14.46 18.41 4.89 49 10 90.02 14 30.01 15.01 19.11 4.71 47 10 90.0O 15 31.07 15.53 19.80 4.55 48 11 90.09 16 32.09 16.04 20.4O 4.45 44 10 89.96 17 33.07 16.54 21. 5 4.28 47 11 Q0.09 18 34.03 17. 2 21.66 4.16 5O 12 90.10 19 34.97 17.48 22.26 4.04 44 11 89.93 20 35.97 17.99 22.86 3.94 48 12 90.07 1 2 3 4 5 6 7 MECHANICS. 175 TABLE II. A NEW MJLL-WRfGf/rS TJItLE, la which the velocity of the whce. is three -sevtntht of the velocity of the watery and the effects of friction on the velocity of the stream reduced to computation. 'S Velocity , of the Velocity of 'he Revolu- tions of Revolu- tionsof i Teeth in the wheel Revolu- tions of . in 1 IJ feet. mill-stone for one of the wheel and staves in the trundle. the mill- stone per minute, by these stavea and teeth. V fi N in S o ^ S2 tj O << Ss Revol. 100 parts of a revol. n -i 8. a > o rt & 2*3 u 5 u H 5 Revol. 100 parts of a revol. 1 7.62 3.27 4.16 21.63 130 6 89.98 2 10.77 4.62 5.88 15.31 92 6 90.02 3 13.20 5.66 7.20 12.50 100 8 9O.OO 4 1.5.24 6.53 8.32 10.81 97 Q 89.94 5 17.04 7.30 9.28 9.70 97 10 90.02 6 18.67 8.00 10.19 8.83 97 11 89.98 7 20,15 8.64 10.99 8.19 90 11 90.01 8 21.56 9.24 11.76 7-65 84 11 89.96 9 22.86 9.80 12.47 7.22 72 10 90.03 10 24. 1O 10.33 13,15 6.84 82 12 89.95 11 25.27 10.83 13.79 6.53 85 13 90.05 12 26.4O 11.31 14.40 6.25 72 12 90.0O 13 27.47 11.77 14.99 6.00 72 12 89.94 J4 28.51 12.22 15.56 5.78 75 13 89.94 15 29.52 12.65 16.13 5.58 67 12 QO.OJ 16 30.48 13.06 16.63 5.41 65 12 89.97 17 31.42 13.46 17.14 5.25 63 12 89.99 18 32.33 13.86 17.65 5.10 61 12 90.01 19 33.22 14.24 18.13 4.96 64 13 89.92 2O 34.17 14.64 18.64 4.83 58 12 89.84 ~ 1 2 3 4 5 Q 176 MECHANICS. Explanation and Use of the Mill-wrights* Tables. It has already been observed, that, according to theory, an undershot wheel will produce the greatest possible effect, when the velocity of the stream is double the velocity of the wheel ; and, upon this principle, the first of the preceding tables has been computed. When we consider, however, that, after every precaution is observed, a small quantity of water will escape between the mill-course and the extremities of the float-boards; and that the effect is diminished by the resistance of the air, and the dispersion of the water carried Theveioci-up by the wheel, the propriety of making the ty of the w heel move with \ of the velocity of the water should, in will readily appear. The Chevalier de Borda sup- practice, be poses j t never to exceed |-, and Mr. Smeaton t-7tnsthat *~ . . , , _ TT . , of the found it to be much nearer \ than y. With 5 therefore, as a proper medium, the numbers in Table II have been computed for this new ve- locity of the wheel. In Table I, the water is sup- posed to move with the same velocity as falling bodies. Owing to its friction on the mill-course, &c. this is not exactly the case; but the error, arising from the neglect of friction, might be in a great measure removed, by diminishing the height of the fall a few inches, in order to have the effec- tive height, with which the other numbers are to be taken out of the table 1 . As this mode of estimating the effects of friction is rather uncer- tain, we have deduced the velocity of the water ~~~ **~ **" *"^* 1 ** " * ' i^ 1 See page 1/0, note. MECHANICS. 177 r om the following formula, F=\/n^x /& \Hh, s 2 in which ^"is the velocity of the water, Rb the Velocity absolute height of the fall, and Hh the depth of ^^ di . the water at the bottom of the course. This for- minished mula is founded on the experiments of Bos&ut, by fnctlofl - from which it appears, that if a canal be inclined -^ part of its length, this additional declivity will restore that velocity to the water which was de- stroyed by friction. We would not advise the mechanic, however, to trust to the second column of Table II for the true velocity of the stream, or to any theoretical results, even when deduced from formula that are most agreeable to experience. Bossut, with great justice, remarks, ' It would not be exact, ' in practice, to compute the velocity of a cur- ( rent from its declivity. This velocity ought to ' be determined by immediate experiment in ' every particular case.' 1 Let the velocity of the water, where it strikes the wheel, be determined by the method which we shall now explain. With this velocity, as an argument, enter column second of either of the tables, according to the velocity which is required for the wheel, and take out the other, numbers from the table,. Method of measuring the Velocity of Water. A variety of methods have been proposed, by Different different philosophers, for measuring the velocity mctho< ! sof r riV t i i n i J ascertain- ot running water. I he method by floating bo- ing the ve- dies, employed by Mariotte ; the bent tube (tube loc *l of _ . . . __ water. ' Traite d'Hydrodynamique, 645. Vol. II. M J78 MECHANICS. recourse) of Pitot ; 3 the regulator of Gugliel- mini; 4 the quadrant, 5 the little wheel, 6 and the method proposed by the Abbe Mann, 7 have each their advantages and disadvantages. The little wheel was employed by Bossut. It is the most convenient mode of determining the superficial velocity of the water ; and when constructed, in the following manner, it will be more accurate, I presume, than any instrument that Plate xn, has hitherto been used. The small wheel IV W - should be formed of the lightest materials. " It should be about 1O or 12 inches in diameter, for mcasur- and furnished with 14 or 16 float-boards. This locity of e ~ wnee l moves upon a delicate screw aB passing running through its axle Bl; and when impelled by the water. stream, it will gradually approach towards _D, each revolution of the wheel corresponding with a thread of the screw. The number of revolu- tions performed, in a given time, are determined upon the scale m a, by means of the index h, fixed at 0, and moveable with the wheel, each division of the scale being equal to the breadth of a thread of the screw, and the extremity h of the index Oh coinciding with the beginning of the scale, when the shoulder b of the wheel is screwed close to the scale a. The parts of a revolution are indicated by the bent index m n pointing to the periphery of the wheel, which is divided into 10O parts. When this instrument is to be used, take it by the handles C\ D, screw the shoulder b of the wheel close to c, so that the indices may both 3 Mem. de 1'Acad. Paris 1732. 4 Aquarum fluentium Mensura, lib. iv. s Bossut,Traite d'Hydrodynamique, 654. 6 Id. Id. 655. " Phil. Tians. v. Ixix, MECHANICS. 17() point to 0, the commencement of the scales ; then, by means of a stop-watch, or a pendulum, find how many revolutions of the wheel are per- formed in a given time. Multiply the mean cir- cumference of the wheel, or the circumference deduced from the mean radius, which is equal to the distance of the centre of impulsion from the axis bB> by the number of revolutions, and the product will be the number of feet which the water moves through in the given time. On ac- count of the friction of the screw, the resistance of the air, and the weight of the wheel, its circum- ference will move with a velocity a little less than that of the stream ; but the diminution of velo- city, arising from these causes, may be estimated with sufficient precision for all the purposes of the practical mechanic. ~ - ON HORIZONTAL MILLS. Although horizontal water wheels are very Horizontal common on the Continent, and are strongly re- milk commended to our notice by the simplicity of their construction, yet they have almost never been erected in this country, and are therefore not de- scribed in any of our treatises on practical me- chanics. In order to supply this defect, and re- commend them to the attention of the mill-wright, we shall give a brief account of their construction. In Fig. 6, we have a representation of one of these Plate i, mills. AE is the large water-wheel, which Flg< 6 ' moves horizontally upon its arbor CD. This arbor passes through the immoveable mill-stone EF at D; and, being fixed to the upper one G//, carries it once round, for every revolution of the great wheel ; JV is the hopper, and /the mill- M2 180 MECHANICS. shoe, the rest of the construction being the same as in vertical corn mills. The mill-course is constructed in the same manner for horizontal as for vertical wheels, with this difference only, that the part mBnC, Fig. 2, of which KL, in Fig. 1 , is a section, instead of being rectilineal like m n, must be circular like TwP, and concentric with the rim of the wheel, sufficient room being left between it and the tips of the float-boards, for the play of the wheel. The equipage 8 of the mill-stone of a hori- zontal mill may be found by multiplying the product of the 100 th part of the expence of the water in cubic feet, and the relative fall, by 5O78, and the product will be the weight of the equi- page in pounds avoirdupois. The mean radius of the wheel A E is to be determined by multiplying the product of the relative fall, and the square root of the expence of water in a second by 0.062. Number, What has been said respecting the number, andform of position, and form of the float-boards, of vertical the float- wheels, may be applied also to horizontal ones. boards. j n ^ j atterj however, the float-boards must be inclined, not only to the radius, but also to the plane of the wheel, with the same angle as they are inclined to the radius, so that the lowest and the outermost sides of the float-boards may be farthest up the stream. velocity of Since the millstone of horizontal mills per- 1 forms the same number of revolutions as the water-wheel ; and since a millstone five feet in diameter should never make less than 48 turns in a minute, the wheel must perform the same 8 The equipage comprehends the mill-stone, the water- wheel, and its arbor. MECHANICS. 181 number of revolutions in the same time ; and in order that the effect may be a maximum, or the greatest possible, the velocity of the current must be double that of the wheel. Suppose the millstone, for example, to be five feet diame- ter, and the water-wheel six feet, it is evident that the millstone and wheel must at least re- volve 48 times in a minute ; and since the cir- cumference of the wheel is 18.8 feet, the float- boards will move through that space in the 48 th part of a minute, that is nearly at the rate of 15 feet per second, which being doubled makes the velocity of the water 30 feet, answering, as appears from the preceding table, to a fall of 14 feet. But if the given fall of water be less than 14 feet, we may procure the same velocity to the millstone by diminishing the diameter of the wheel. If the wheel, for instance, is only five feet diameter, its circumference will be 15.7 feet, and its floats will move at the rate of 1 2.56 feet in a second, the double of which is 25. 1 2 feet per second, which answers to a head of wa- ter less than ten feet high. As the diameter of the water-wheel, however, should never be less than seven times the breadth of the mill-course at K, (Fig. l), there will be a certain height of the fall beneath which we cannot employ hori- zontal wheels, 9 without making the millstone re- volve too slowly. This height will be found by the following table. 9 This applies only to mills for grinding corn, where the millstone is fixed on the arbor of the water-wheel, and must move with a determinate velocity. For any other purpose horizontal wheels may be used, however small be the fall of water. M 3 MECHANICS. Method of of finding whether horizontal or vertical mills should be erected. When the natural depth of the water at the bottom of the fall is to the breadth of the mill- course at the same 3 to 1 2 to 1 1 to 1 Equal. -vtol ytOl place, as The relative fall beneath which we cannot employ ho- rizontal mills, will be 7.314 8.602 11.350 14.976 17.613 Ft. Dee Ft. Dec. Ft. Dec Ft. Dec Ft. Dec. Perform- ance of ho. rizontal will*. Thus, if the natural depth of the water at K 9 Fig. 1, is three times the width of the mill- course at the same place, the relative fall be- neath which we cannot employ a horizontal wheel will be 7-314 feet. Since the depth of the water is so great in this case, a great quan- tity of it will be discharged in a second, and therefore it requires a less velocity, or a less height of the fall, to impel the wheel, whereas if the depth of the water had been only one third of the width of the mill-course, such a small quantity would be discharged in a second that we must make up for the want of water by giving a great velocity to what we have, or by making the height of the fall 17.613 feet. In order to find the radius of the millstone in horizontal mills, multiply the expence of water in cubic feet in a second, by the relative fall ; extract the square root of the product, and mul- tiply this root by O.267, the product will be the radius of the millstone in feet. The quantity of meal ground in an hour may be found by the rules already given; for vertical mills, or by multiplying the product of the ex- MECHANICS, 183 pence of water, and the relative fall, by 456ft, and the result will be the quantity required. The thickness of the millstone at the centre and circumference, the thickness of the arbor and pivots, may be determined by the rules al- ready laid down for vertical mills. In horizontal wheels, the mill-course is some- Horizontal times differently constructed. Instead of the2* water assuming a horizontal direction before it float- strikes the wheel, as in the case of undershot- boards> mills, the float-board is so inclined as to receive the impulse perpendicularly, and in the direction of the declivity of the waterfall. When this construction is adopted, the greatest effect will be produced when the velocity of the float-boards is not less than ^ sin~3? ' * n w ^.ich H repre- sents the height of the waterfall, and A the angle which the direction of the fall makes with a vertical line. But since this quantity increases as the sine of A decreases, it follows, that with- out taking from the effect of these wheels, we may diminish the angle A, and thus augment con- siderably the velocity of the float-boards, accord- ing to the nature of the machinery employed ; whereas, in vertical wheels, there is only one determinate velocity, which produces a maximum effect. 5 In the southern provinces of France, where With curvi horizontal wheels are very generally employed, j the float-boards are made of a curvilineal form, so as to be concave towards the stream. The Chevalier de Borda observes, that in theory a double effect is produced when the float-boards 1 See Memoire sur les Roues Hydrauliques, Mem. de L'Acad. Royal. Par. 1767, p. 285. 184 MECHANICS. are concave, but that this effect is diminished IB practice, from the difficulty of making the fluid enter and leave the curve in a proper direction. Notwithstanding this difficulty, however, and other defects which might be pointed out, hori- zontal wheels with concave float-boards are ah ways superior to those in which the float-boards are plain, and even to vertical wheels, when there is a sufficient head of water. When the float-boards are plain, the wheel is driven merely by the impulse of the stream ; but when they are concave, a part of the water acts by its weight, and increases the velocity of the wheel. If the fall of water be five or six feet, a horizon- tal wheel with concave float-boards may be erect- ed, whose maximum effect will be to that of ordinary vertical wheels as 3 to 2. Conical ho- j n the provinces of Guyenne and Languedoc, wheel a w ith an ther species of horizontal wheels is employed spiral float, for turning machinery. They consist of an in- verted cone, AE^ with spiral float-boards of a xin.Fig.a.curvilineal form winding round its surface. The A PP' wheel moves on a vertical axis in the building^ D ), and is driven chiefly by the impulse of the water conveyed by the canal C to the oblique float-boards. When the water has spent its impul- sive force, it descends along the spirals, and continues to act by its weight till it reaches the bottom, where it is carried off by the canal M. ON DOUBLE CORN MILLS. Double It frequently happens that one water-wheel s ' drives two millstones, in which case the mill is said to be double ; and when there is a copious discharge of water from a high fall, the same MECHANICS. 1S.> water-wheel may give sufficient velocity to three, ibur, or five millstones. Mr. Ferguson has given a brief description of a double mill in vol. i, p. 101, and a drawing of one in Plate VII, Fig. 4, but has laid down no maxim of con- struction for the use of the practical mechanic. In supplying this defect, let us first attend to double horizontal mills, in which the axis CD, Fig. 6, is furnished with a wheel which gives ** motion to two trundles, the arbors of which carry the millstones. In order to find the weight of the equipage for each millsone, multiply the product of the expence of water, and the relative fall, by 48116ft, and divide the product by 2000, if there are two millstones, 3OOO if there are three, and so on, the quotient will be the weight of the equipage of each millstone. To determine the radius of the wheel that size of the drives the trundles, find first the radius of the J r ^| JJJ^ millstones by the rules already given, and having trundles. added it to half the distance between the two neighbouring millstones,* subtract from this sum the radius of the lantern, which may be tak- en at pleasure, and the remainder will be the radius required when there are two millstones. But if there are three millstones, or four, or five, or six, before subtracting the radius of the lantern, divide the sum by 0.864, 0.705, O.587, O.5, respectively. The mean radius of the water-wheel may besreeofthe found by multiplying the square root of the re- lative fall by the radius of the millstone, by the ^r* This quantity may be taken at pleasure, and should never be less than 2 feet, however great be the number of Ac millstones. 18G MECHANICS. radius of the wheel that drives the trundles, and by 231, and then dividing the product by the radius of the lantern multiplied by 1000, the quotient will be the wheel's radius. It may hap- pen, however, that the diameter of the wheel found in this way is too great. When this is the case, we may be certain that the radius of the lantern has been taken too small. In order then to get a less value for the wheel's radius, increase a little the radius of the lantern, and find new numbers both for the water-wheel, and that which drives the trundles, by the preceding rule. It may happen also, that in giving an ar- bitrary value to the radius of the lantern, the diameter of the wheel found by the rule may be too small, that is, less than seven times the breadth of the mill-course at the bottom of the fall. When this takes place, make the diameter of the water-wheel seven times the width of the mill-course, and you may find the radius of the other wheel and lanterns by the following rules, size of the 1 . To find the radius of the wheel that impels wheel that t ^ e trunc ]i es . ^^ t h e radius of the millstone drives the, , 1 . . 7 . .... trundle, to hair the distance between any two adjoining millstones for a first quantity. Multiply the square root of the relative fall by the radius of the millstone and by .231 ; and having divided the product by the radius of the water-wheel, add unity to the quotient, and multiply the sum by 1 if there are two millstones, by ,8(J4 if there are three, by .705 if there are four, by .587 if there are five, and by .5 if there are six, and the result will be a second quantity. Divide the first by the second quantity, and the quo- tient will be the radius of the wheel that drives the trundles. 2. To find the radius of the lantern, multiply MECHANICS. 187 the radius of the wheel as found by the pre- size of th ceding rule, by the square root of the relative 1 * 3 fall, and by .231, and divide the product by the radius of the water-wheel, the quotient will be the lantern's radius. By the rules formerly given find the quantity of meal ground by one millstone, and having multiplied this by the number of millstones, the result will be the quantity ground by the com- pound mill. If the equipage of the millstone of a vertical mill, as found in p. 159, should be too great, that is, if it should require too large a millstone, then we must employ a double mill, like that which is represented in Plate VII, Fig. 4, or one which has more than two millstones. In order to know the equipage of each mill- stone, find it by the rule for a single mill, and having multiplied the quantity by .947, divide the product by the number of millstones, and the quotient will be the equipage of each millstone. The radius of the wheel Z>, Plate VII, Fig. 4, will be found by the same rule which was given for horizontal mills ; but it must be attended to, that the lantern whose radius is there employed isnot#, but^G, or EH. To determine the mean radius of the large size of the spur-wheel AA^ which is fixed upon the arbor 8 P ur - whcel - of the water-wheel, multiply the square of the radius of the lanterns F G or E //, by the radius of the water-wheel, and also by 43O2, and a first quantity will be had. Multiply the square root of the relative fall by the radius of one of the millstones, and by the radius of the wheel Z>, and by 1OOO, and a second quantity will be ob- tained. Divide the first quantity by the second, and the quotient will be the mean radius of the wheel A A. 18S MECHANICS. The quantity of meal ground by a compound mill of this kind, is found by the same rule that was employed for compound mills driven by a horizontal water-wheel. ON BREAST MILLS. Breast ^ breast water-wheel partakes of the nature both of an overshot and an undershot wheel : it ' is driven partly by the impulse, but chiefly by the weight of the water. Fig. 1 , of Plate II, re- presents a water-wheel of this description, where MC is the stream of water falling upon the float- board o } with a velocity corresponding to the height in , and afterwards acting by its weight upon the float-boards between o and B. The mill-course oB is concentric with the wheel, which is fitted to it in such a manner that very little water is permitted to escape at the sides and extremities of the float-boards. The effect of a mill driven in this manner, is equal, accord- ing to Mr. Smeaton, ' to the effect of an under- * shot mill, whose head is equal to the differ- * ence of level between the surface of water in * the reservoir and the point where it strikes the ' wheel, added to that of an overshot, whose ' height is equal to the difference of level be- ' tween the point where it strikes the wheel and s the level of the tail water.' 3 That is, the effect of the wheel A is equal to that of an undershot wheel driven by a fall of water equal to m n, added to that of an overshot wheel whose height is equal to n D. M. Lambert of the Academy Smeaton on Mills, Scholium, p. 36. MECHANICS. 189 of Sciences at Berlin, 4 has shewn, that when the float-boards arrive at the position op, they should be horizontal, or the point p should be lower than o, in order that the whole space between any two adjacent float-boards may be filled with water; and that C m should be equal to the depth of the float-boards. He observes also, that a breast wheel should be used when the fall of water is above four feet, and below ten, provid- ed the discharge of water is sufficiently copious ; that an undershot wheel should be preferred when the fall is below four feet, and an overshot wheel when the fall exceeds ten feet ; and that, when the fall exceeds 12 feet, it should be divid- ed into two, and two breast mills erected. This, however, is only a general rule which many cir- cumstances may render it necessary to overlook. The following table, which may be of essential utility to the practical mechanic, is calculated from the formulae of Lambert, and exhibits at one view the result of his investigations. * Nouv. Mem. de 1'Acad. de Berlin, 1775, p. 7J. 190 MECHANICS. Height of the fall in feet.= CZ>, Fig. i, Plate II. o o o o o <> w - to e IN overshot mills, where the wheel is moved power of k t ^ e we jorj lt o f fae wa er m the buckets, each water on / i-rr i i overshot bucket has a different power to turn the wheel ; wheels. anc j ^jg p 0wer i s proportioned to the distance of the bucket from the top or bottom of the wheel ; or more accurately, to the sine of the arch con- tained between the centre of the bucket and the top or bottom of the wheel, according as the bucket is above or below its centre. The bucket, for instance, placed upon the top of the wheel, has no power to turn it ; the bucket next to this contributes but a little to turn the wheel, because it is virtually placed at the extremity of a very short lever ; whereas the bucket, which is equal- ly distant from the top and bottom of the wheel, and which is level with the centre, has the great- est power to turn it, because it acts at the extre- mity of a lever equal to the wheel's semidiameter. If we suppose, then, that each bucket contains MECHANICS. 193 One gallon of water, equal in weight to 10.2 ft avoirdupois ; we may, by the simplest oper- ations in trigonometry, compute, in pounds avoir- dupois, the power which each bucket exerts in turning the wheel ; and, by taking the sum of these, we will have the effective weight of the water 3 in the buckets, and, consequently, its pro- portion to the real weight of the water, with which the semi-circumference of the wheel is loaded. Those who choose to make this calcu- lation, will find that the total weight of water upon the semi-circumference of an overshot wheel is to the effective weight as 1 to .637 ; but, as two or three of the buckets at the bottom of the arch are always empty, the proportion will rather be as 1 to .75 nearly. From these principles, we may deduce the following method, simpler than any hitherto given, of computing the effec- tive weight of water upon overshot wheels of any diameter. i,i -,Is} }? i>\lm v.jq 'iiM^/ -T srioi RULE. Multiply the constant number 6.12 by half the number of buckets, and this pro- finding duct by the number of gallons in each bucket, and the result will be the effective weight of the water upon the wheel, three buckets being sup- posed empty at the bottom. This rule is pretty accurate for wheels from 20 to 32 feet in dia- meter. But when the diameter of the wheel is ' This phrase, which is used by practical mechanics, is very exceptionable ; as every drop of water in the buckets, excepting the vertical bucket, is effective. By the effective weight of the water, therefore, we must understand that weight which, if suspended at the opposite extremity of the wheel, would keep it in equilibria or balance the loaded arch. J' r ol II N 194 MECHANICS, less than 20 feet, the answer given by the rule must be diminished one pound avoirdupois for every foot which the wheel is less than 2O. Suppose that it is required to find the effective weight of water upon a wheel 1 8 feet in diameter, having 4O buckets, each containing two gallons ale measure. Then 6.12x20x2=244.8. But as the diameter of the wheel is two feet below 2O, we must deduct two pounds from the pre- ceding answer, and the result will be 242.8 ft avoirdupois. On the performance of Overshot and Undershot Mills* a*.-Inl'inritl . -fi C.''"" Perform- From a number of accurate experiments made- overshtt by tne ingenious Mr. Fenwick, upon a variety of and under- excellent overshot mills, it appears, that when shotwheels -the water wheel is 20 feet in diameter, 392 gal- lons of water per minute (ale measure) will grind one boll of corn per hour (Winchester measure); 675 gallons per minute will grind 2 bolls ; 94 gallons will grind 3 bolls j 1270 gallons will grind 4 bolls, and 1623 gallons will grind 5 bolls. From these data it will be easy to com- pute the performance of an overshot mill, what- ever be the diameter of the wheel and the supply of water. Exampls. Ex AMPLE 1. Let it be required to find how many bolls of corn will be ground by an over- shot mill, driven by a wheel 25 feet in diameter, upon which 1 1 50 gallons of water are discharg- ed in a minute. Say, as the nearest number Galls. Bolls. Galls. Bolls. 1270 : 4=1150 : 3.62, the quantity of com MECHANICS. 195 ground by a wheel 20 feet in diameter. Then to find the quantity which a 25 feet wheel will Feet. Bolls. Feet. Bolls. grind, say, as 20 : 3.62i=25 : 4.52 the answer required. EXAMPLE 2. If it is required to grind 3^- bolls of corn per hour, where the stream discharges gallons in a minute, what must be the diameter 2220 of the wheel ? Find the number of gallons which a 20 feet wheel will require for grinding the given quantity of corn by the following propor- Bolls. Galls. Bolls. Galls. tion. As 4 : 12703.5 : 1 1 1 1. Then, by inverse Galls. Feet. Galls. Feet. proportion, 1 J 11 : 2On:2220: 1O the diameter of the wheel required. In order to find the quantity of corn ground Perform- by an undershot mill, which is moved by a simi- an " of . . J . ' _ * undershot lar wheel, and a similar quantity or water, as an mills, overshot mill ; divide the quantity ground in an overshot mill by 2.4, and the quotient will be the answer. If it is required to know what size of wheel is necessary for making an undershot mill grind a certain quantity of corn, the supply of water being given ; find the size of an overshot wheel necessary for producing the same effect, and multiply this by 2.4; the product will be rhe required diameter of the undershot wheel. N2 96 MECHANICS. On the formation of the Buckets, and the pmper Velocity of Overshot Wheels. Plate in, Let AM (Plate III, Fig. 4) be part of the ri g-4- shrouding, or ring, of buckets of an overshot Form of wheel, GOFABCD is the form of one of these the buckets buckets. The shoulder, AB, of the bucket Wheels" 110 ' Sh uld be OIle half f AE * the de P th f the shrouding; AF should be y more than AE. The arm, BC, of the bucket must be so inclined to AE, that HC may be of AE; and CD, the wrist of the bucket, must make such an angle with BCn, the direction of the arm, that Dn may be | of ;?. improve- A very considerable improvement, in the con- Mr" Burns, struction of the bucket, has been made, by Mr. Robert Burns, at Cartside, Renfrewshire. He divides the bucket, by a partition, mB, of such a height, that the portions of the bucket, on each side of it, may be of equal capacity. Dr. Robi- son observes, that this principle is susceptible of considerable extension, and recommends two or more partitions, particularly when the wheel is made of iron. By this means, the fluid is retain- ed longer in the lower buckets, and when there is a small supply of water, it may be delivered into the outer portion of the bucket, which, be- ing at a greater distance from the centre of mo- tion, increases the power of the water to turn the wheel. Dr. Robison advises, that the rim of the wheel, and consequently the breadth of the buckets, should be pretty large, in order that the quantity of water, which they receive from the spout, may not nearly fill the bucket. The spout, MECHANICS. 197 which conveys the water, should be considerably narrower than the breadth of the bucket ; and the shoulder AE should be perforated with a few holes, in order to prevent the water from being lifted up by the ascending buckets. The distance of the spout, from the receiving bucket, should, hi general, be two or three inches, that the water may be delivered with a velocity a little greater than that of the rim of the wheel ; otherwise the wheel will be retarded, by the impulse of the buckets against the stream, and much power would be lost, by the water dashing over them. 5 The proper velocity of an overshot wheel is a Velocity of point, upon which some celebrated mechanicians have entertained different sentiments. From a variety of experiments, Mr. Smeaton infers, in general, that the circumference of the wheel should move with the velocity of a little more than three feet per second. ' Experience,' says he, ' confirms, that this velocity of three feet ' in a second is applicable to the highest over- ' shot wheels, as well as the lowest ; and all ' other parts of the work being properly adapted * thereto, will produce, very nearly, the greatest * effect possible ; however, this also is certain, ' from experience, that high wheels may deviate ' farther from this rule, before they will lose J If the spout be one inch and seven-tenths above the re? Height of ceiving bucket, it will deliver the water with the velocity t}lc 8 P ut of the wheel, that is about three feet per second. In order, ab ve . the therefore, to make the velocity of the water exceed a little that of the wheel, the height of the spout should be 2-J- inches, and the water will move at the rate of three feet seven inches per second. ' Dr. Robison recommends three or four inches ; but this is evidently too great, as four inches gives a velocity of four feet seven inches per second. N3 1Q8 MECHANICS. ' their power, by a given aliquot part of the * whole, than low ones can be admitted to do; for ' a wheel of 24 feet high may move at the rate ' of six feet per second, without losing any con- . * siderable part of its power j and, on the other * hand, I have seen a wheel of 33 feet high, that * has moved very steadily and well with a velocity c but little exceeding two feet/ 6 M. Deparcieux ' shews, that most work is Depar- performed by an overshot mill, when it moves slowly, and that the more we retard its mo- tion, by increasing the work to be performed, the greater will be the performance of the wheel. This important conclusion was deduced, by The effect Deparcieux, from experiments made upon a small of overshot , r , .. r t i i ^ whaeh in- wheel, 20 inches in diameter, furnished with 48 verseiy as buckets, which received the water like a breast- city. '" wheel. On the axis of this wheel were placed cylinders of different sizes, the smallest being one inch, and the largest four inches, in dia- meter, around which was wrapped a cord, with a weight attached to it. * When the one-inch cy- linder was used, a weight of 1 2 ounces was ele- vated to the height of 69 inches and 9 lines; and a weight of 24 ounces was elevated 40 inches. When the four-inch cylinder was em- ployed, a weight of 12 ounces was raised to the altitude of 87 inches and 9 lines, and a weight of 24 ounces to the height of 45 inches and 3 lines. From these results, it is evident, that, with the ' Sjmeaton on Mills, p. 33. 1 Mem. de 1'Acad. Paris, 1754, p. GO3, 6/1, 4 to . ; p. 928, 1033rS T0 . 1 The model employed, in Mr. Smeaton's experiments, resembles very much that of Deparcieux, though their e.x periments were made about the same time. MECHANICS, 191) Tour-inch cylinder, when the motion was slowest , the effect was greatest, and that, when a double weight was used, which diminished the wheel's velocity, the weight was raised to more than half its former height. J This increase of performance, by diminishing causes of the wheel's velocity, has been ascribed to differ- this * cnt causes. Deparcieux and Brisson account for it, by saying, that when the motion of the wheel is slow, the same portion of water acts more ef- ficaciously. Dr. Robison and Mr. Smeaton ascribe it to a greater quantity of water pressing on the wheel ; for, when the wheel's motion is slow, the buckets receive more water as they pass the spout. One of the most powerful causes, however, is, a diminution of the centrifugal force 1 Mr. J. Albert Euler, whose memoir, on the best Me- thod of employing the Force of Water and other Fluids, gained the prize, proposed by the Royal Society of Got- tingen, in 1754, has also shewn, that the more slowly an overshot, or breast-wheel, with buckets, moves, the greater will be its performance. See the Comment. Getting. 1/54, or the Journal Stranger, Dec. 1756. Mr. Smeaton, too, deduces, from his experiments, this general rule, that, c&r ter'u paribui, the less the velocity of the wlieel, the greater will be its effect. But he obferves, on the other hand, that, when the wheel of his model made about 20 turns in a minute, the effect was nearly the greatest; when it made 3O turns, the effect was diminished about -^ part; and that, when it made 40, it was diminished about one-fourth ; when it made less than 1 S turns, its motion was irregular ; and when it was loaded so as not to admit its making 18 turns, the wheel was overpowered by its load ; Smeaton on Mills, p. 33. For farther information on this subject, we must refer the reader to the original memoirs quoted above, in the first of which Deparcieux proves his point, by rea- soning, and in the second, by experiment ; or to Brisson's Traite de Physique, torn, i, p. 306, edit. 3, where there is a general view of Deparcieux's experiments. 20O .MECHANICS. of the water in the buckets ; for, when the velo- city of the wheel is great, the water, receding from the centre, is thrown out of the buckets, and they are emptied sooner than they would have been, had the wheel moved with less velo- city. * n t ^ ie Memoirs f tne Academy of Berlin, for wheels. 1755, M. Lambert has published a dissertation on the theory of overshot mills ; but does not seem to be in the least degree acquainted with the improvements, which have been made upon them, in this country. He supposes the bucket? Plate in, to have the form GFfg, (Fig. 4, Plate III), so Flg> * that about one quarter only of the circumference of the wheel is filled with water. Notwithstand- ing these circumstances, however, the following table, computed from his formulas, may be of cqnsiderable advantage to the millwright. MECHANICS. H>* 1 > 1 1 to H- o co QO -si 7 r itllfl i Ji'frl- 5*l*f* _ ... r : U ic ^) r n 50 n ? ^ 2 ^ a. r 1 s- to 00 - O O5 fcO 00 O5 tn ^ W lO CO O o n O *" M ^fc r^s-^gi % o n CO O5 ^O W Oi >- Ot wi O5 a- > i' ri? ir?^s ??&. O O O O O O I? 2^3 3" . CO Ox Cn t^ *> W CO n 2* 3^ ^ CO -* 00 ^- Cn 00 fcO Crt O n p s*'-*^^ =- 5 t I V 1 O5 Oi , the position of the ' s ' 3 ' float-boards. In common undershot wheels, their motion is greatly retarded by the resistance op- posed by the tail water to the ascending float- boards ; and their velocity is still farther diminish- ed by the resistance of the air. But when the preceding construction is adopted, the resistance of the air and the tail water is greatly diminished by the oblique position of the float-boards. Account of an Improvement in Flour Mills. In most of the flour mills in Scotland and Eng- improve- land, a considerable quantity of manual labour is " necessary before the wheat is converted into flour. When the grain is ground and conveyed into the trough from the mill stones, it is afterwards put into bags and raised to the top of the mill house, 204- MECHANICS, to be laid into the cooling boxes or benches, from which it is conveyed into the bolting ma- chine., to be separated from the bran or husk. This manual labour may be saved by adopting an improvement, for which we are indebted -to the ingenuity of the American mill-wrights. A large screw is placed horizontally in the trough, which receives the flour from the millstones. The thread or spiral line of the screw is compos- ed of pieces of wood about two inches broad, and three long, fixed into a wooden cylinder 7 or 8 feet in length, which forms the axis of the screw. When the screw is turned round this axis, it forces the meal from one end of the trough to the other, where it falls into another trough, from which it is raised to the top of the mill house by means of elevators, a piece of ma- chinery similar to the chain pump. These ele- vators consist of a chain of buckets or concave vessels like large teacups, fixed at proper dis- tances upon a leathern band, which goes round two wheels, one of which is placed at the top of the mill house, and the other at the bottom, in the meal trough. When the wheels are put in motion, the banjl revolves, and the buckets, dip- ping into the meal trough, convey the flour to the upper storey, where they discharge their contents. The band of buckets is inclosed in two square boxes, in order to keep them clean, an.d preserve them from injury. f III --'r . r MECHANICS. ; ! - ON DR. BARKER'S MILL. t VMi-2i '! THIS mill, which is sometimes called Parent's improvc- M7/, has already been described at considerable^^ " length in the supplement. ' It has exercised the mill. ingenuity of Euler and Bernouilli ; and its theory seems to be as complicated as its construction is simple. Instead of conveying the water into the top of the vertical pipe DB, M. Mathon de la Cour* proposes to bend the pipe A, which con- piate m, ducts the water from the reservoir, down by the Su P iFi *' 1 - letters ONGP^ and to introduce the fluid into the horizontal arm C at the point g. When the water is thus conveyed into the machine, it rushes out at the holes d and e, with a velocity corres- ponding to the height of the reservoir, and the trunck c will revolve with a retrograde motion. The cause of this is obvious. If the hole d were shut up with a cylindrical pin, the pressure upon the circular area of its base would be equal to a column of water whose base is equal to this cir- cular area, and whose length is the height of the water in the reservoir. But the same force is 1 See Rozier*s Journal de Physique, Jan. and Aug. 1775. 6ee . 97 of this volume. 206 MECHANICS. Mode of its exerted on an equal portion of the tube opp6site " to the aperture d. The pressures, therefore, up- on each side of the arm c will be equal and op- posite, and no motion will ensue. As soon, how- ever, as the hole d is opened, the pressure is re- moved from that part of the tube, and the arm c is driven backwards by the unbalanced pressure on the opposite side. Dr. Robison imagines that this unbalanced pressure is equal to the weight of a column of water, having the orifice for its base, and twice the depth under the surface of the water in the trunk for its height, upon the supposition that the arm C is also impelled by the reaction of the issuing fluid. Upon this sup- position, which is extremely plausible, Barker's mill must be a very powerful machine ; and when water is used as the impelling power of machinery, it will produce much greater effects by its reaction, than it does either by its impulse; or gravity. The effect As soon as the machine begins to move, the "d by'the " horizontal arm withdraws itself from the pres- sure ; the impelling power is consequently dimi- nished, as it depends upon the relative velocity of the arm C, and the issuing fluid. Dr. Desa- guliers 3 maintains, that when the engine is in mo- tion, the pressure is equal to the weight of a column, which would make the velocity of efflux equal to the relative velocity of the fluid and the machine ; and from this he concludes, that it will produce a maximum effect when the veloci- ty of the arm is y of the velocity acquired by falling from the surface of the water in the reser- voir, in which case it will raise to the same height Experimental Philosophy, vol. ii, 3 a Edit. p. 45.0* MECHANICS. 207 JL of the water expended, though T * T is the quan- tity raised by an undershot mill. 'The velocity of the machine is no doubt in- , , t r i r r i inertia of creased by the centrifugal force or the water in t he fluid, the arms ; but this effect is completely counter- acted by the inertia of the fluid. For, as a new quantity of water is constantly entering the arms, a considerable portion of its velocity must be lost in communicating to this water the circular mo- tion of the arms. This diminution of velocity may be prevented in some measure by enlarging the diameter of the horizontal arms, which will cause the water to move more slowly to the aperture ; but when this is to be done, the form Form re- recommended by Euler will be most advantage- S'bJSur. ous. In Fig. 4, is represented a section of the P i ate xni> machine, with this form. The canal a delivers the App.Rg./v. water into the bason CDMN, in the direction of the tangent, and with the same velocity as the machine. The water then descends in spiral ex- cavations formed by partitions between the co-, noids CF, EM, and DE,FN; and when it reaches the bottom F 9 the water flows off in the direction of the tangent, by means of a spout for each excavation. It has often occurred to me, that a very power- New kin* ful hydraulic machine might be constructed, by^ h * e * tcr ~ combining the impulse with the reaction of water. S uggeitco formed as to convey the water easily into the spiral excavations, we should have a machine something like the conical horizontal wheel in Fig. 2, with spiral channels instead of spiral float-boards, and which would in some measure be moved both by the impulse, weight, and reaction of the water. MECHANICS* Practical Rules for the Construction of Barker's Mill, given by Mr. Waring.* tactical i ^ Make the arm of the rotatory tube, from the centre of motion to the centre of the aper- ture, of any convenient length not less than -5- ( according to Mr. Gregory, 5 who has correct- ed some of Waring's numbers) of the perpendi- cular height of the water's surface above their centres. 2, Multiply the length of the arm in feet by .614, and take the square root of the product for the proper time of a revolution in seconds, and adapt the other parts of the machinery to this velocity ; or, 3, If the time of a revolution be given, mul- tiply the square of this time by 1.63 for the pro- portional length of the arm. 4, Multiply together the breadth, depth, and velocity, per second, of the race, and divide the last product by 18.47 (14.27 according to Mr. Gregory) times the square root of the height, for the area of either aperture. 5, Multiply the area of either aperture by the height of the head of water, and the product by 41y (55.775 according to Mr. Gregory) pounds, foi- the moving force estimated at the centres of the apertures in pounds avoirdupois. 6, The power and velocity at the aperture may be easily reduced to any part of the machinery by the simplest mechanical rules. 4 Transactions of the Americ. Phil. Soc. vol. iii, p. 1 93. * Mechanics, vol. ii, p. 111. MECHANICS. 209 M. Mathon de la Cour gives us the following Dimensions dimensions of one of Dr. Barker's mills, which f a ^ r , s f was erected at Bourg Argental, in order to work mills. ventilators for a large room. The length of thepiate in, horizontal trunk C was 7 feet 8 inches, and its Fi s- x ' Su P t diameter 3 inches ; the diameter of the orifices, at d and e, 1 inches ; the height of the reser- voir, above the trunk C, is 21 feet ; the diameter of the pipe, which conveyed the water into C, from below, was 2 inches at their junction, and was fitted into it by grinding. When this machine was doing no work, and when the fluid issued only from one aperture, it performed 115 revolutions in a minute. The aperture, therefore, was moving with the velocity of 46 feet per second ; whereas, if this aperture were at rest, the water would have issued only with a velocity of 3 7 feet per second. Dr. Robison l observes, that this great velocity was produ- ced by the prodigious centrifugal force of the water. Might it not be advantageous to have another horizontal arm crossing C at right angles ? J * Encyclop. Britan. vol. xviii, p. 909, where the reader will find some excellent remarks upon this machine. J Those, who wish to study this important subject with attention, will find the investigations of Euler in the Mem. Acad. Berlin, 1751, and in the Nov. Comment. Petrop. torn, vii, and those of Bernouilli, at the end of his Hydrau- lics. See also the Exercitationes Hydraulicae of Professor Segner, who gives Barker's mill as an invention of his own ! and the woiks which have already been quoted. J. A. Euler proposed a machine to be driven by the re-action of the water in the Com. Getting. 1755. L o MECHANICS. ON THE FORMATION OF THE TEETH OF WHEELS AND THE LEAVES OF PINIONS. matLn of' A HOUGH nothing is more essential to the per- the teeth of fection of machinery than the proper formation wheels, &c ri iri_i Ji. r ' or the teeth or wheels, and those parts of en- gines, by which their force and velocity are conveyed to other parts; yet no branch of mechanical science has been more overlooked by the speculative and practical mechanics of this country. In vain do we search our systems of experimental philosophy for information on this point. Their authors seem either to reckon it beneath their notice, or to be unacquainted with thelabours of De la Hire, Camus, and other foreign academicians, who have written very ingenious dissertations on the teeth of wheels. It is in the memoirs, indeed, of these philosophers, that all our knowledge upon this subject is contained, if MECHANICS. 211 we except a few general remarks, by the learned Mr. Robison. 3 It would be easy to shew, did the nature .of this work permit, that when one wheel drives duced by another, it is not driven with an uniform force jj l t c e y e J ld ~ and velocity, or, in other words, the one wheel will act sometimes with greater, and sometimes with less, force, and the other will move some- times with a greater, and sometimes with a less velocity, unless the teeth of one or of both the wheels be parts of a curve, generated after the manner of an epicycloid, 4 by the revolution of another curve along the convex or concave side of a circle. It will be sufficient, for our pre- sent purpose, to shew, that, when one wheel impels another, by the action of epicycloidal teeth, the moments of these wheels will be equal. Let the wheel B Fig. 7, drive the platel - wheel A^ by the action of the epicycloidal teeth Fl *' 7> m n, m 1 ;/, &c. upon the infinitely small pins, or spindles, a, , c, and let the epicycloids, mn 9 &c. be generated, by the circumference c b a, moving over the circumference m" m' m. It is evident, from the formation of the epicycloid, that the arch a b is equal to the arch m m', and 5 Two ingenious memoirs have also been written upon this subject, by A. G. Kaestner, entitled, De Dentibus Rotarum, and published in the Comment. Reg. Soc. Dot- ting, vol. iv, and v, 1/81, &c. The celebrated Eider has also treated this subject with great ability, in his memoir De aptiisima Flgura Rotarum Dentibus tribucnda, Nov. Com- ment. Petrofol. 1754, 1755, torn, v, p. 299. 4 Under curves, of this description, are comprehended those which are formed, by evolving the circumferences of circles, for it is demonstrable, that these involutes arc epicycloids, the centres of whose generating circles are in- finitely distant. 02 MECHANICS* the arch a c to m m" ; for, when the pa*t I m^ of the epicycloid m' n f , is forming, every point of the arch a b is applied to every point of the arch m m! ; and the same may be said of the arch a c. Since, then, the wheels B and A 9 that is, the power and the weight, move through equal spaces, in equal times, equal weights act- ing in opposite directions, at the points a and w, will be in equilibrio: but, as the power of the wheel B must always be greater than the resistance of the wheel A^ which is put in motion,, this power will, during the whole of the action, have the same relation to the resist- ance which it overcomes, and the one wheel will impel the other with an uniform force and velocity. Th pro- For the discovery of this property of the epi- r[cydoid 1C cycloid, which Dr. Robison erroneously ascribes discovered to De la Hire, or Dr. Hook, we are indebted to by Rocmer. t j le D an i s h astronomer Glaus Roemcr, the dis- coverer of the progressive motion of light ; and Wolfius, upon whose authority this fact is stated, 7 laments, that the mechanics of his time did not avail themselves of the discovery. In order to insure an uniformity of pressure and velocity, in the action of one wheel upon another, it is not necessary that the teeth, either of one or both wheels be exactly epicycloids. 7 Ex eodem fonte Ofaus Ro-merus, cum Parisiis commo- raretur, quamvis non sine subsidio Geometriue sublimioris, deduxit figuram dcntium in rotis epicycloidalem esse de- here ; id quod post eum quoqiie ostendit Phtlippus de la Hire: sed quod dolendumhactenus injpraxin recepta non est. Wolfii Opera Mathemat. torn, i, p. 684. The same fact is stated by Leibnitz, in the Miscellan. Bcrolinens. p;. 315. MECHANICS. 213 if the teeth of one of them be either circular, or triangular, with plain sides, or like a triangle, with its sides converging to the wheels' centre, or, indeed, of any other form, this uniformity of force and motion will be attained, provided that the teeth of the other wheel have a figure which is compounded of that of an epi- cycloid and the figure of the teeth of the first wheel. 8 But, as it is often difficult to describe this compound curve, and sometimes im\ ossJble to discover its nature, we shall endeavour to select such a form for the teeth as may be easily described by the practical mechanic, while it ensures an uniformity of pressure and velocity. But, in order to avoid circumlocution and ob- scurity, we shall call the small wheel, (which is supposed always to be driven by a greater one), the pinion, and its teeth the leaves of the pinion. The line, which joins the centres of the wheel and pinion, is called the line of centres. Now there are three different ways, in which the teeth of one wheel may -act upon the teeth of another; and each of these modes of action requires a differ- ent form for the teeth. I. When the teeth of the wheel begin to act upon Different the leaves of the pinion, just as they arrive at m des > in , i . f f , , / . , which one the line or centres; and when their mutual wheel may action is carried on after they have passed this act u P on line. anoher - II. When the teeth of the wheel begin to act upon the leaves of the pinion, before they * M. de la Hire has shewn, in a variety of cases, how to find this compound curve. 3 214- MECHANICS. arrive at the line of centres, and conduct them either to this line or a very little be- yond it. III. When the teeth of the wheel begin to act upon the leaves of the pinion, before they ar- rive at the line of the centres, and continue to act after they have passed this line. First mode J. The first of these modes of action is of action, recommen( j e( j by Camus and De la Hire, the latter of whom has investigated the form of the teeth solely for this particular case. It is Plate n, represented, in Fig. 2, where B is the centre Ig ' a< of the wheel, A the centre of the pinion, and AE the line of centres. It is evident, from the figure, that the part b of the tooth a b of the wheel, does not begin to act on the leave m of the pinion, till they arrive at the line of centres AB; and that all the action is carried on after they have passed this line, and is completed when the leaf m comes into the situation rz, 4 When this mode of action is adopted, the acting faces of the leaves of the pinion should be parts of an interior epicycloid generated by a circle, of any diameter, rolling upon the con- cave superfices of 'the pinion, or within the circle a d h ; and the acting faces a b of the teeth of the wheel should be portions of an ex- terior epicycloid, formed by the same generating circle, rolling upon the convex superfices o dp of the wheel. Now, it it is demonstrable, that when one circle rolls within another, whose diameter is 4 The tooth c, in the Figure, should have been in con- tact with the leaf , and pn the point of quitting it. MECHANICS, 215 double that of the rolling circle, the line gene- rated, by any point of the latter, will be a straight Jine, tending to the centre of the larger circle. If the generating circle, therefore, men- A straight tioned above, should be taken, with its diameter, line ma 7 be equal to the radius of the pinion, and be made to roll upon the concave superficies m b h of the all y- pinion, it will generate a straight line, tending to the pinion's centre, which will be the form of the acting faces of its leaves, and the teeth of the wheel will, in this case, be exterior epicycloids, formed by a generating circle, whose diameter is equal to the radius of the pi- nion, rolling upon the convex superfices odp of the wheel. This form of the teeth, viz. when the acting faces of the pinion's leaves are right lines, tending to its centre, is exhibited, in Fig. 3, and is, perhaps, the most advantage- Fi s- 3- ous, as it requires less trouble, and may be executed with greater accuracy than if the cur- vilineal form had been employed. It is recom- mended, both by De la Hire and Camus, as par- ticularly advantageous in clock and watch work. The attentive reader will perceive, from Fig. 2, Fig.^. that, in order to prevent the teeth of the wheel from acting upon the leaves of the pinion, before they reach the line of centres AB ; and that one tooth of the wheel may not quit the leaf of the pinion, till the succeeding tooth begins to act upon the succeeding leaf, there must be a certain proportion between the number of leaves in the pinion, and the number of teeth in the wheel ; or between the radius of the pinion, and the radius of the wheel, when the distance of the leaves AE is given. But, in machinery, the number of leaves and teeth is always known, from the velocity, which is required at the work- 216 MECHANICS. ing point of the machine. It becomes a matter, therefore, of great importance, to determine, with accuracy, the relative radii of the wheel and pinion. 9 Relative di- For this purpose, let ^, Fig. 3, be the thTwhee / pinion, having the acting faces of its leaves and pinion, straight lines, tending to the centre, and B the centre of the wheel, AB will be the distance of their centres. Then, as the tooth C is sup- posed not to act upon the leaf Am, till it arrives at the line AB, it ought not to quit Am, till the following teeth F has reached the line AB. But, since the tooth always acts in the direction of a line drawn perpendicular to the face of the leaf Am, from the point of contact, the line CH, drawn at right angles to the face of the leaf Am, will determine the extremity of the tooth CD, or the last part of it, which should act upon the leaf Am, and will also mark out CD, for the depth of the tooth. Now, in order to find AH, HB, and CD, put a for the number of teeth in the wheel, b for the number of leaves in the pinion, c for the distance of the pivots A and B, and let x represent the radius of the wheel, and y that of the pinion. Then, since the circumference of the \\ heel is to the circumference of the pinion, as the number of teeth in the one to the number of leaves in the other, and as the circumferences of circles are proportional to their radii, we shall have % a : /; ,r : y, then, by composition, (Eucl. v. 18), a-{-b : b~c : y, (c being equal to x-\-y\ and, c A very ingenious Proportion-compass has been invent- ed, by M. le Cerf, watchmaker, at Geneva, for finding the relative diameters of wheels and pinions. It is desert <"" at length in the Phil, Trans. T. 6y, p. 50. MECHANICS. 21? consequently, the radius of the pinion, viz. c b ' y -r-j then, by inverting the first analogy, ^~ we have b : aiz.y : x, and, consequently, the radius of the wheel, viz. x -7^, ij being now a known number. Now, in the triangle AHC, right angled at C, the side AH is known, and likewise all the 36O angles (HAC being equal to -y-) the side AC^ therefore, can be easily found by plain trigono- metry. Then, in the oblique angled triangle ACB, the angle CAB, equal to HAC, is known, and also the two sides A B, AC^ which contain it; the third side, therefore, viz. C B, may be determined ; from which D B, equal to H B, already found, being subtracted, there will re- main CD for the depth of the teeth. When the action is carried on after the line of centres, it often happens that the teeth will not work in the hollows of the leaves. In order to prevent this, the angle C B H must always be greater than half the angle HBP. The angle HBP is equal to 360 degrees, divided by the number of teeth in the wheel, and C BH is easily found by plain trigonometry. (See p. 228.) Instead of pinions or small wheels, the mill- Teeth of wrights in this country frequently substitute Ian- terns or trundles, which consist of cylindrical staves, fixed at both ends into two round pieces of board. From the use of truncllts, however, Dr. Robison discourages the practical mechanic, when he observes, that De la Hire justly con- * demned the common practice of making the ' small wheel or pinion in the form of a Ian- * tern,' and that, when ' the teeth of the large 218 MECHANICS. ' wheel take a deep hold of the cylindrical pms ' of the trundle, the line of action is so disad- f vantageously placed that the one wheel has * scarcely any tendency to turn the other.' 5 It is with the greatest deference to such an able philosopher as Dr. Robison, that we presume to contradict this statement, both with respect to the fact which is asserted, and the principle which is maintained. In no part of De la Hire's Dis- sertation upon this subject does he condemn the use of lanterns. On the contrary, he actually demonstrates, that when the teeth of the great wheel are formed in a particular manner, and drive a small wheel whose teeth are cylindrical pins, the pressure and angular velocity of the one wheel will be equal to the pressure and an- gular velocity of the other ; or, in other words, their action will be uniform. To this form of the teeth of the great wheel, when those of the small wheel are cylindrical, we shall now direct the reader's attention ; and we earnestly recom- mend it to the notice of the practical mechanic, because it furnishes us with a method of remov- ing, or at least of greatly diminishing, the friction which arises from the mutual action of the teeth. Method of Let A, Fig. 4, be the centre of the pinioa cur^ep^ or sma N wheel TCH 9 whose teeth are circular raiiei to an like ICR, having their centres in the circle p'tlr^n PDE - Upon ^ the centre f the Iarge whee1 ' Fig* 4? ' at ^e distances B C, B D 9 describe the circles FCK, GDO ; and with PDE, as a generating circle, form the exterior epicycloid DNM, by rolling it upon the convex superficies of the cir- cle GDO. The epicycloid DNM thus formed, J The same observation is made in Imison's Elements of Science and Art. Vol. i, p. 91. MECHANICS. 219 would have been the proper form for the teeth of the large wheel GDO, had tlie circular teeth of the small wheel been infinitely small ; but as their diameter must be considerable, the teeth of the wheel should have another form. In order to determine their proper figure, divide the epi- cycloid DNM into a number of equal parts, 1, 2, 3, 4, &c. as shewn in the figure, and let these divisions be as small as possible. Then, upon the points 1 , 2, 3, &c. as centres, with the distance D C, equal to the radius of the circular tooth, describe portions of circles similar to those in the figure ; and the curve OPT, which touches these circles, and is parallel to the epicycloid DNM, will be the proper form for the teeth of the large wheel. In order that the teeth may not act upon each other till they reach the line of centres A B, the curve P should not touch the circular tooth ICR till the point has arrived at D. The tooth P, therefore, will commence its action upon the circular tooth at the point /, where it is cut by the circle D R E. On this account, the part ICR of the cylindrical pin being super- fluous, may be cut off, and the teeth of the small wheel will be segments of circles similar to the shaded parts of the figure. But if the spindles remain entire, the vacuities between the teeth should be cut out, and their sides OK directed to the centre of the wheel. If the teeth of wheels and the leaves of pi- nions be formed according to the directions al- ready given, they will act upon each other, not only with uniform force, but also with very little fric- Vci 7 lk don. The one tooth rolls upon the other, and [weenie" neither slides nor rubs to such a degree as to teeth when retard the wheels, or wear their teeth. But as p , rope l ly . . . .. , . . . r shaped. it is impossible in practice to give that perfect t ^ 22O MECHANICS. curvature to the acting faces of the teeth which theory requires, a certain quantity of friction will remain after every precaution has been taken in the formation of the communicating parts. This friction may be removed, or at least greatly diminished, by the following contrivance. If, instead of fixing the circular teeth, as in Fig. 4, to the wheel DRE^ they are made to move upon axles or spindles fixed in the circum- ference of the wheel, all the friction will be taken away, except that which arises from the motion of the cylindrical tooth upon its axis, The ad- vantages which attend this mode of construction Cylindrical are many and obvious. The cylindrical teeth teeth mov- i r 11 11*11 ing on their ma y be formed by a turning lathe with the great- axes, est accuracy ; the curve required for the teeth of the large wheel is easily traced ; the pressure and motion of the wheels will be uniform ; and the teeth are not subject to wear, because what- ever friction remains is almost wholly removed by the revolution of the cylindrical spokes about their axis. The reader will also observe, that this improvement may be most easily introduced when the small wheel has the form of a trundle or lantern ; and that it may be adopted in cases where lanterns could not be conveniently used. PLATE in, In Fig. 3, is represented the manner by which r g- 3- cylindrical teeth, moveable upon their axis, may be inserted in the circumference of wheels. B is the part of the wheel on which the tooth is to be fixed ; A is the cylindrical tooth which moves upon its axis b c made of iron, whose extremi- ties run in bushes of brass, fixed in the project- ing pieces of wood b, c. This improvement, however, can only be adopted where the ma- chinery is large. For small works, the teeth of the pinion or small wheel should be rectilineal s and those of the large wheel epicycloidal. MECHANICS; IL Having hitherto supposed i that the mutual second action of the teeth does not commence till theyjjjj^ arrive at the line of centres, let us now attend to the form which must be given them, when, the whole of the action is carried on before they reach the line of centres, or when it is com- pleted a very little below this line. This 'mode of action is not so advantageous as that which we have been considering, and should, if pos- sible* always be avoided. It is represented *%]?"" nf> Fig. 1, where A is the centre of the pinion, S that of the wheel, and AB the line of centres. It is evident from the figure, that the tooth C of the wheel acts upon the leaf D of the pinion be- fore they arrive at the line BA; that it quits the leaf when they reach this line, and have assumed the position of E and F; and that the tooth c works deeper and deepeV between the leaves of the pinion the nearer it comes to the line of cen- tres. From this last circumstance a considerable quantity of friction arises, because the tooth C does not, as before, roll upon the leaf D, but slides upon it ; and from the same cause the pi- nion soon becomes foul, as the dust which lies upon the acting faces of the leaves is pushed into the hollows between them. One advantage, however, attends this mode of action, far it al- lows us to make the teeth of the large wheel rectilineal, and thus renders the labour of the mechanic less, and the accuracy of his work greater, than if they had been of a curvilineal form. If the teeth C, E, therefore, of the wheel B C are made rectilineal, having their surfaces directed to the wheel's centre, the acting . faces of the leaves Z>, F 9 &c. must be epicycloids formed by a generating circle, whose diameter is equal to the radius Bo of the circle c>/>, roll- MECHANICS. ing upon the circumference m n of the pinion A. But if the teeth of the wheel and the leaves of , the pinion are made curvilineal, as in the figure a the acting faces of the teeth of the wheel must be portions of an interior epicycloid formed by any generating circle rolling within the concave superficies of the circle op, and the acting faces of the pinion's leaves must be portions of an ex- terior epicycloid, produced by rolling the same generating circle upon the convex circumference m n of the pinion. When the teeth of the large wheel are cylin- drical spindles, either fixed or moveable upon their axis, an exterior epicycloid must be pi ATE n, formed, like DNM, in Fig. 4, by a generating *' s ' 4 ' circle whose radius is AC, rolling upon the con- vex circumference FCK, AC being in this case the diameter of the wheel, and FCK the cir- cumference of the pinion. By means of this epicycloid a curve OPT must be formed as be- fore described, which will be the proper curva- ture for the acting faces of the leaves of the pi- nion, when the teeth of the wheel are cylindrical, though, when this is the case, this mode of action ought to be avoided. The relative diameter of the wheel and pinion, when the number of teeth in each is known, may be found by the same formulae which were given for the first mode of action, with this dif- ference only, that in this case the radius of the wheel is reckoned from its centre to the extre- Third m ity o f its teeth, and the radius of the pinion mode or r i i r \ action. " om ts centre to the bottom or its leaves. III. The third way in which one wheel may drive another, is when the action is partly carried on before the acting teeth arrive at the line of centres, and partly after they have passed this line. This mode of action, which is represented in MECHANICS. 223 "Fig. 2, is a combination of the two first modes, PLATE ir> and consequently partakes of the advanta- Fi - * ges and disadvantages of each. It is evident from the figure, that the portion eh of the tooth acts upon the part b c of the leaf till they reach the line of centres A B, and that the part e d of the tooth acts upon the portion b a of the leaf after they have passed this line. It follows, therefore, that the acting parts e h and b c must be formed according to the directions given for the first mode of action, and that the remaining parts ed, la, must have that curvature which the second mode of action requires ; consequent- ly e h should be part of an interior epicycloid formed by any generating circle rolling on the concave circumference m n of the wheel, and the corresponding part b c of the leaf should be part of an exterior epicycloid formed by the same ge- nerating circle rolling upon /; E 0, the convex circumference of the pinion : the remaining part c d of the tooth should be a portion of an exte- rior epicycloid, engendered by any generating circle rolling upon e L, the concave superficies of the wheel ; and the corresponding part b a of the leaf should be part of an interior epicycloid described by the same generating circle, rolling along the concave side b E of the pinion. As it would be extremely troublesome, however, to give this double curvature to the acting faces of the teeth, it will be proper to use a generating circle, whose diameter is equal to the radius of the wheel B C, for describing the interior epicy- cloid e h and the exterior one b c, and a gener- ating circle, whose diameter is equal to A - the arch ES, or angle EAS, being equal to -j- -> and CS, or the angle CAS, being equal to - their difference EC, or the angle EAC, will be , 36O 360 X -a u^ i_ equal to --- 7^- By subtracting, we have 360 ml 360 In j j* j* u ; - -r-; - , and dividing by b, gives 360m 360 a 36OXw n rpi , T-, ^^ j , or Y * he angle EA C cm bm being thus found, the triangle] EAB 9 or 1AB, which is almost equal to it, is known, because AB is given ; and likewise AI t which is equal to the cosine of the angle IAB, AC being ra- dius, and A 1C being a right angle j consequently IB the radius of the wheel may be found by tri- gonometry. It was formerly shewn, 3 that AC 9 the radius of what is called the primitive pinion, was equal to ^7, and that EC, the radiu? of the primitive wheel, was equal to j-?. If, thert, we subtract A Cor AS from AP, we shall have the quantity SP, which must be added to the ra- dius of the primitive pinion ; and if we take the J Sec page 217. n. P 226 MECHANICS. difference of BC (or BL) and DE, the quantity LE will be found, which must be added to the radius of the primitive wheel. We have all along supposed that the wheel drives the pinion, and have given the proper form of the teeth upon this supposition. But when the pinion drives the wheel, the form which was given to the teeth of the wheel, in the first cae, must in this be given to the leaves of the pinion ; and the shape which was formerly given to the leaves of the pi- nion must now be transferred to the teeth of the wheel. Form of Another form for the teeth of wheels, different according from any which we have mentioned, has been re- to Dr. B.O- commended by Dr. Robison. He shews that a perfect uniformity of action may be secured, by making the acting faces of the teeth involutes of PLATE iv, the wheel's circumference. Thus, in Plate IV, Fig. 1 , let AB be a portion of the wheel on which the tooth is to be fixed, and let Ap a be a thread lap- ped round its circumference, having a loop hole at its extremity a. In this loop hole fix the pin a, and with it describe the curve or involute abcdeh, by unlapping the thread gradually from the circumference Ap m. This curve will be the proper form for the teeth of a wheel, whose dia- meter is AB. Dr. Robison observes, that as this form admits of several teeth to be acting at the same time, (twice the number that can be admif- ted in M. de la Hire's method), the pressure is divided among several teeth, and the quantity upon any one of them is so diminished, that those dints and impressions, which they unavoidably make upon each other, are partly prevented. He candidly allows, however, that the teeth thus formed are not altogether free from sliding and friction, but that this slide is so insignificant as MECHANICS. 22? to amount only to -^ of an inch, when a tooth three inches long, fixed on a wheel ten feet in diameter, acts upon the teeth of another wheel whose diameter is two feet. It may be proper to observe, that this form of the teeth which Dr. Robison recommends is not new. It is only a modification of the general principle of De la Hire, ' that no curve is proper ' for the teeth of wheels, unless it be epicycloidal,' i. e. generated after the manner of an epicy- cloid. A straight line can be generated by an epicycloidal motion ; and the involute a b c, &c. is actually an exterior epicycloid,* whose base is Ap m S t and the centre of whose generating circle is infinitely distant. The involute a b c d, Mechanic. &c. may also be produced by an epicycloidal nio-^^?^ tion ; for, since the circumference of a general- ing inv- ing circle, whose centre is infinitely distant, must lutcs - be a straight line, we may form the involute a b c, by making a straight ruler roll upon the circumference of the circle to be evolved. In Fig. 1 , let on be a straight ruler at whose ex- tremity is fixed the pin n 9 and let the point of the pin be placed upon the point m of the circle, then by rolling the straight ruler upon the cir- cular base, so that the point in which it touches the circle may move gradually from m towards B, the curve m n will be generated exactly simi- lar to the involute a b c, &c. This, by the by, is perhaps an easier and more accurate method of generating involutes than by unlapping a thread from the circumference of the evolute or circle to be evolved. ir ~'rn^-"! .-."i T * >"> ~ : 4 De la Hire calls involutes the last of the exterior epi- cycloids. P2 228 MECHANICS. From what has been said in page 217, the reader will perceive, that when the pinion has a small number of leaves, the first mode of action cannot be employed. By computing the angles PLATE H ff B C, HBP, Plate II, Fig. 3, trigonometri- cally, it will be found that a pinion of seven leaves cannot be impelled uniformly by a wheel of fifty teeth, when the action is carried on after the line of centres ; for even if the leaves had no breadth like a mathematical line, then there would be no room left for the play of the teeth. However great indeed be the number of teeth in a wheel, the space between them would not be sufficiently great to receive a leaf of a rea- sonable thickness, and to leave at the same time a sufficient space for the play of the teeth. The same may be said of a pinion of 8 leaves driven by a wheel of 57 teeth and upwards, and of a pinion of 9 leaves driven by a wheel of 64" teeth and upwards. When a pinion therefore of 7, 8, or 9 leaves are to be impelled by a wheel, the action of the teeth upon the leaves must com- mence before they have reached the lines of cen- tres, and be continued after they have passed that line. If the pinion has ten leaves, it may be moved uniformly by a wheel of 72 teeth, by the first mode of action ; but if the vacuity be- tween the teeth is equal to, or greater than the teeth themselves, the leaves of the pinion must be caught by the teeth a little before they reach the line of centre. s 4 The number of teeth here specified are those beneath which it would be impossible for the action to be carried on. When the wheel has a greater number, there is no impossibility in the case ; but the teeth would be too sl&i- der to resist the strains to which they are exposed. 5 See Camus's Cours de Mathematique, 550. MECHANICS. 229 Thus have we endeavoured to lay before our readers all the information which we have upon this important subject ; and we trust it will be candily received, as it is the only essay on the subject which has appeared in our language. 6 The demonstrations of the propositions have been purposely left out, as being rather foreign to the object of a practical work. To the mechanic, they are of no consequence ; and the mathema- ' In a book entitled, fmison's Elements of Science and Art t which professes to be a second edition of Imison's School of Arts, there are some practical directions for the formation of the teeth of wheels, but they are so defective in prin- ciple that they cannot be trusted. The author seems merely to have heard that the acting faces of the teeth should be cpicycloidal, but to have been totally ignorant whether the epicycloids should be exterior or interior, and what should be their bases and generating circles. The directions which this author gives for forming the teeth of a rack, and the lifting cogs or cams of forge hammers, are equally desti- tute of scientific principle. ** The preceding note, which was published in the first edition of this work, has called forth a reply from the author of the article in Imison's Elements, on which I had animadverted. This reply was published in a translation of one of Camus's Essays on the teeth of wheels, and was writ- ten by a Mr. Thomas Gill in London. This gentleman in- sists, that the rules which he has given in Imison's dementi are correct, because they have been found to answer in practice ; though it is demonstrable, and evident to every person who understands the subject, that the generating circles with which lie describes his epicycloids, are twice as large as they ought to be. The same gentleman has thought proper to say, that the preceding artkle on the teeth of wheels is defective, in not containing that method of form ing the teeth in which the acting faces are partly radii, and partly epicycloids ; while thit very method is not only given, but recommended, to the notice of the practical me- chanic ! (See page 223, line 11 from bottom). I sh.ll forbear making any animadversions on this new mode of literary assault. I willingly commit the subject to the judgment of every intelligent reader. P3 23O MECHANICS. tician can either demonstrate them himself, of have recourse to the original dissertations of Ca- mus and De la Hire. On the Nature of BEVELLED WHEELS, and the method of giving an epicucloidal form to their rr *L 1 eeth. Nature of The principle of bevelled wheels was pointed wheeii d out by ^ e k Hire, so long ago as the end of the seventeenth century. 7 It consists in one fluted or toothed cone acting upon another, as represented FLATExn.in Figure 8, of Plate XII, where the cone OD Fig. 8. drives the cone C, conveying its motion in the direction C. If these cones be cut parallel to their bases as at A and J3, and if the two small cones between A B and be removed, the re- maining parts A C and B D may be considered as two bevelled wheels, and B D will act upon A C in the very same manner, and with the same effect that the whole cone D acted upon the whole cone C ; and if the section be made nearer the bases of the cones, the same effect will be produced. This is the case in Figure 9, Fig. 9 . where CD and D E are but very small portions of the imaginary cones A CD and AD E. In order to convey motion in any given di- rection, and determine the relative size and situ- ation of the wheels for this purpose, let A B, Fig. 10. Fig. 1O, be the axis of a wheel, and CD the given direction in which it is required to convey the motion by means of a wheel fixed upon the axis A B, and acting upon another wheel fixed on the axis CD, and let us suppose that the axis C D must have four times the velocity of A B, fc. ; eo gfto - $iT~~ 7 Traite de Mecanique, prop. 66, published in the Menu de 1'Acad. Paris, &c. depuis 1666, jusqua 1699, torn. ix. MECHANICS. 231 or must perform Four revolutions while A B per* forms one; then the number of teeth in the wheel fixed upon A B must be four times great- er than the number of teeth in the wheel fixed upon CD, and their radii must have the same proportion. Draw c d parallel to C D at any convenient distance, and draw a /' parallel to A B at four times that distance, then the lines i m and in drawn perpendicular to AzB and CD respect- ively, will mark the situation and size of the wheels required. In this case the cones are n i and m i, and s r n /, rp m ?, are the portions of them that are employed. The operation here indicated, is evidently nothing more than the common problem of dividing a given angle BOD into two parts, whose sines shall be to one another as the number of revolutions of the one wheel, to the number of revolutions of the other. If m : n as one of these numbers is to the other, the problem solved algebraically will give the following theorem. 2. Sin. - =: 2 Sin. ~sr X 2* Jt ^ , which verifies the preceding construction. 8 m -j- ' The formation of the teeth of bevelled wheels On the for- is more difficult than one would at first imagine ; and no author, so far as I know, has attempted to direct the labours of the mechanic. The teeth of such wheels, indeed, must be formed by the same rules which we have given for other wheels ; but since different parts of the same tooth are at different distances from the axis, these parts must have the curvature of their act- ing surfaces proportioned to that distance. Thus, in Fig. 9, the part of the tooth at i must be 8 See Gregory's Mechanics, vol. ii, p. /, and Simpson'* Select Exercises, p. 138. 232 MECHANICS, more incurvated than the part at C, as is evident from the inspection of Fig. 8, and the epicycloid for the part i Fig. 9. must be formed by means of circles whose diameters are i m and Ff 9 while the epicycloid for the part r must be generated by circles, whose diameters are Cn and D d. Let us suppose a plane to pass through the points A B ; the lines A B, A 0, will evident- ly be in this plane, which may be called the Plane of Centres. Now, when the teeth of the wheel D E, which is supposed to drive C Z), the smallest of the two, commence their action on the teeth of CD, as soon as they arrive at the plane of centr.es, and continue their action after they have passed this plane, the curve given to the teeth of CDtt.C should be a portion of an interior epicy- cloid formed by any generating circle rolling on the concave superficies of a circle whose diameter is twice C n, which is perpendicular to C A, and the curvature of the teeth at i should be part of a similar epicycloid, formed upon a circle, whose diameter is twice i m. The curvature of the teeth of the wheel Z) at Z) should be part of an ex- terior epicycloid formed by the same generating circle rolling upon the concave circumference of a circle whose diameter is twice D d, perpendicu- lar to D A; and the epicycloid for the teeth at F is formed in the same way, only instead of twice D d, the diameter of the circle must be twice Ff, When any other mode of action is adopted, tht? teeth are to be formed in the same manner that we have pointed out for common wheels, with this difference only, that different epicycloids are necessary for the parts F and D. It may be suf- ficient, however, to find the form of the teeth at F, as the remaining part of the tooth may be shaped by directing a straight rule from different MECHANICS. 233 points of the epicycloid at F to the centre A, and filing the tooth till every part of its acting sur- face coincides with the side of the ruler. The reason of this operation will be obvious by at- tending to the shape of the tooth in Fig. 8. When the small wheel CD impels the large one D E, Fl s-*- the epicycloids which were formerly given to CD must be given to DE, and those which were given to D E must be transferred to C D. 1 The wheel represented in Fig. 5, Plate XIII, On crown is sometimes called a crown wheel, though it i s whed evident from the figure, that it belongs to that species of wheels which we have just been con- sidering ; for the acting surfaces of the teeth PLATE both of the wheel MB, and of the pinion*" 1 *. EDG, are directed to C, the common vertex of the two cones C M Z?, C E G. In this case, the rules for bevelled wheels must be adopted, in which AS is to be considered as the radius of the wheel for the profile of the tooth at A, and M N as its radius for the profile of the tooth at M\ and the epicycloids thus formed, will be the sections or profiles of the teeth in the direc- tion M P, at right angles to MC, the surface of the cone. When the vertex C of the cone MCG approaches to N till it be in the same plane with the points M, G 9 some of the curves will be cycloids, and others involutes, as in the case of rackwork mentioned in page 242 ; for then the cone C E G will revolve upon a plane sur- face. 1 A method of bevelling wheels with a simple instru- ment, invented by Mr. James Kelly of New Lanark cotton- mills, maybe seen in the Repertory of Arts, vol. vi, p. 1O6. This instrument, founded on the equality of vertical angles, is so very simple that it must have occurred to mechanics of common ingenuity. 234 MECHANICS. On the Formation of Exterior and Interior Epi- cycloids^ and on the Disposition of the Teetk on the Wheels Circumference. Mchanicai Nothing can be of greater importance to the format f P ract ical mechanic, than to have a method of epicycloids, drawing epicycloids with facility and accuracy ; the following, we trust, is the most simple me- chanical method that has yet been devised. PLATE iv, Take a piece of plain wood G //, Fig. 2, and fix Flg ' *' upon it another piece of wood , having its cir- cumference mb of the same curvature as the cir- cular base upon which the generating circle AB jj is to roll : when the generating circle is large, the shaded segment B will be sufficient. In any part of the circumference of this segment, fix a sharp pointed nail a, sloping in such a manner that the distance of its point from the centre of the circle may be exactly equal to its radius j and fasten to the board GHz piece of thin brass, or copper, or tinplate a b, distinguished by the dot- ted lines. Place the segment B in such a posi- tion that the point of the nail a may be upon the point by and roll the segment towards G, so that the nail a may rise gradually, and the point of contact between the two circular segments may advance towards m ; the curve a b described upon the brass plate will be an accurate exterior epicy- cloid. In order to prevent the segments from sliding, their peripheries should be rubbed with rosin or chalk ; or a number of small iron points may be fixed in the circumference of the generat- ing segment. Remove, with a file, the part of the brass on the left hand of the epicycloid, and the remaining concave arch or gage a b will be a pattern tooth, by means of which all the rest may be easily formed. When an interior epicycloid is MECHANICS. 235 wanted, the concave side of its circular base must be used. The method of describing it is repre- sented in Figure 3, where CD is the generating Fig. 3, circle, F the concave circular base, M N the piece of wood on which this base is fixed, and c d the interior epicycloid formed upon the plate of brass, by rolling the generating circle C, or the generating segment D, towards the right hand. The cycloid, which is useful in forming the teeth of rack work, is generated precisely in the same manner, with this difference only, that the base on which the generating circle rolls must be a, straight line. Although, in general, it is necessary to give Both sides the proper curvature only to one side of theJ^JJj^JJ* teeth, yet it may be proper to form both sides properly with equal care, that the wheels may be able to shap 2 b X sin. ^-we have T VI, r, 7 i >TIO JL^^ Loganthm 2 .b =. 1.7125636 Log.Sine^orlOzr9.2396702 m Therefore OZ^S.QSS*, 0.9522338 The radius of curvature at the point D, con- sequently, will be= x OAwhenz=1 * The radius of curvature being always = r ' - ' w s h will be equal, in the present example, to orlxaOAorX OD. MECHANICS. 239 is, the radius of curvature will be 13.030. If z be successively diminished to 6, 4, and 2, we shall have the results contained in the following table, which are found in the same way as when z X EBO BOD BO OD Radius of Curvature. 2 3 20' 6 40 1 2 2 4O 7 5 20 1.50O6 3.0007 1.5004 2.9996 2.1824 4.3631 6 1O 3 8 O 4.50OO 4.4962 6.54OO 12 12 6 16 O 8.9876 8.9584 13.O300 By means of this table, four points of the epicy- cloid may be found. Make the angle BA 1 2, EOD=l6, and OD = 8.9584, which will deter- mine the point D ; and so on with the rest. As it would be extremely difficult to project the wheels C and A upon paper, when they are very large, we shall shew how to describe the epicy- cloid without using the centres C and A. Draw BE perpendicular to the line CA that joins the centres of the wheels, and make the angle EBO equal to one half of z, viz. 6 degrees. Make B 0, as before found, equal to 8.9876; the angle BOD=\6, and OZ>=8.9584, and the point D will be determined when the line CA\s only given in position. In the Cycloid let the line B (Fig. 6, Plate PLATE IIT, III) be equal to b X z, b being the radius of the Fi s- 6 ' generating circle c, and z any number of degrees taken at pleasure. Then )0~2^x Sine J,and m DOB-. From Diet fall the perpendicular DK, and let DK=y t and SK=x ; thenZJoX 2 sine - DK,ory=2b x sine- =Z> X versed sine 24O MECHANICS. z= X 1 cos. z. Likewise we have KO2 b x cos.- Xsine ^=b X sine z. Whence B K 9 or x~b 2 ^ X z sine z. Wherefore jB A" and D K being thus found, the point D in the cycloid will be determined ; and by diminishing z continually, we shall then have other points of the cycloid be- tween D and , and by increasing it we shall have points beyond D. Example. To illustrate this by an example, let /Jz: 1 and z =110 = 1 80 60, then since/; zz 1 we shall have y versed sine 120 2 versed sine 60 1.500. To find JT, which is 6 x z sine z, or, in the present case, ~ z sine z, since b is equal to 1 . The arch z, or 1 20, being of the circumference of a circle whose radius is 1 , and whose circum- ference is 3.1415927 X 2, or 6.2831854, will be equal to 2.0943950, and the sines of 1 20, or its supplement 60, is 0.8660254. Therefore 12Or=2.0943951 sine 120 O.866O254 x= 1.2283697 If z be made 5, x will be zzO.OOOl 108, and#= O.OO38053. The numbers x and y being thus determined, we have only to make BK equal to jr, and KD to y, in order to find the point D. It may be proper to observe, that the variable number z should be taken pretty small both for the cycloid and epicycloid, as it is only a little por- tion of these curves that is required for the teeth of wheels ; and when several points of the curve are determined, the intervening space may be made arches of a circle equicurve to the epicy- cloid at the same point. 5 9 9 See Kaestner's Memoir de Dentlbis Rotarum, in the Cpmment. Sac. Reg. Getting. 1782, vol. v, pp. 9, 24 rMflj ."i. -., -i j . MECHANICS, . *V 'IO e ':,j0 ;\l~> I!' flOJU .'/ C L)'J ON THE FORMATION OF THE TEETH OF RACK- WORK, THE ARMS OF LEVERS, THE WIPERS OF STAMPERS, AND THE LIFTING COGS OR CAMS OF FORGE HAMMERS. 1 ) i*s .'JO .'"JIT??! ,i''iii , !.'._ i j-ij THE teeth of a wheel may act upon those of a Q rack, according to the three different ways which one wheel acts upon another, and each of these modes of action requires a different form for the communicating parts. From what has been said, in the preceding dis- sertation, it would be easy to deduce the proper form for the teeth of rack\vork, merely by con- sidering the rack as part of a wheel, whose centre is infinitely distant. 8 But, as the epicycloids are, in this case, converted into other curves, which have different names, and are generated in a dif- ferent manner, it may be proper, for the sake of 8 Since this article was written, 1 have seen a paper, by Kaestner, on the same subject, in the Nov. Commcntar. Soc. Reg. Gottinf. 1771, torn, ii, p. 1 17) but it contains nothing new, excepting a method of describing involutes, by means of points. Pol. IL Q 242 MECHANICS. the practical mechanic, to add a few observations 1 illustrative of this subject. plate iv, i n pig. 4 ? i e t AE be the wheel, which is em- 1 ' 4 ' ployed to elevate the rack C 9 and let their mutual action not commence till the acting teeth have reached the line of centres AC. In this case, C becomes, as it were, the pinion or wheel driven, and the acting faces of its teeth must be interior epicycloids, formed by any generating circle, roll- ing within the circumference pq ; but as pq is a straight line, these interior epicycloids will be cycloids, or trochoids, as they are sometimes call- ed, which are curves generated by a point in the circumference of a circle, rolling upon a straight line, or plane surface. The acting face op, therefore, will be part of a cycloid, formed by any generating circle, and mn the acting face of the teeth of the wheel, must be an exterior epi- cycloid, produced by the same generating circle, rolling on mr the convex surface of the wheel. If it is required to make op a straight line, as in the figure, then mn must be an involute of the circle mr, formed according to the manner re- presented in Fig. 1, Plate IV. Figure 4 represents likewise a wheel depressing the rack C when the third mode of action is used, viz. when the action commences above the line of centres, and is carried on below this line. In this case also, C becomes the pinion, and DE the wheel ; eh, therefore, must be part of an interior epicycloid, formed by any generating circle, roll- ing on the concave side ex of the wheel, and be must be an exterior epicycloid, produced by the same generating circle, rolling upon the circum- ference of the rack. The remaining part cd of the teeth of the wheel must be an exterior epi- cycloid, described by any generating circle, mov- MECHANICS. 243 ing upon the convex side ex, and ba must be an interior epicycloid, engendered by the same ge- nerating circle, rolling within the circumference of the rack. But, as the circumference of the rack is, in this case, a straight line, the exterior epicycloid be, and the interior one ba, will be cy- cloids, formed by the same generating circles which are employed in describing the other epi- cycloids. Since it would be difficult, however, as has already been remarked, to give this com- pound curvature to the teeth of the wheel and rack, we may use a generating circle, whose dia- meter is equal to Z).v, the radius of the whee, for describing the interior epicycloid eh, and the exterior one be, and a generating circle, whose diameter is equal to the radius of the rack s for describing the interior epicycloid ab, and the ex- terior one de; ab and eh, therefore, will be straight lines, and be will be a cycloid, and de an invo- lute of the circle ex, the radius of the rack being infinitely great. In the same manner may the form of the teeth of rackwork be determined, when the second mode of action is employed, and when the teeth of the wheel, or rack, are circular or rectili- neal. But, if the rack be part of a circle, it must have the same form for its teeth, as that of a wheel of the same diameter with the circle, of which it is a part. In machinery, where large weights are to be importance raised, such as in fulling-mills, mills for pounding: / giving o i ' r , . o the proper ore, &c. or where large pistons are to be ele-f or m to vated by the arms of levers, it is of the greatest *'P C - consequence, that the power should raise the weight with an uniform force and velocity ; and this can be effected only by giving a proper form to the wipers. A certain class of mechanics ge- Q2 244 MECHANICS. nerally excuse themselves for not attending to the proper form of the teeth of wheels, by alleging that the scientific form differs but little from theirs, and that teeth, however badly formed, will, in the course of time, work into the proper shape. This excuse, however, will not apologize for their negligence in the present case. The scientific form of the wipers of stampers and the arms of levers are so widely different from the form which is generally assigned them, as to increase very much the performance of the machine, and preserve its parts from that injury which is always occasioned by the want of an uniform motion. Now there are two cases in which this uni- formity of motion may be required, and each of these demands a different form for the commu- nicating parts. 1 , When the lever is to be rais- ed perpendicularly, as the piston of a pump, &c.; , 2, When the lever to be raised or depressed moves upon a centre, and rises or falls in the arch of a circle, in the same plane with the wheel, such as the sledge hammer in a forge, and the stampers in a fulling mill ; and 3, When the lever rises in the arch of a circle, and moves in a plane at right angles to the plane of the wheel's motion. Plate iv, i. In Figure 5, let AE be an axis driven Flg> 5 ' by a water- wheel, or any other power, at ^gjj t l r ^ es right angles to which is fixed the bar mm, on perpendi- whose extremities the wipers mn mn are fastened, cuiariy. rj-^g w jp er mn ^ ac ting upon the arm PE, raises the piston, or weight, EF to the required height. The piston then falls, and is again raised by the. lower wiper. We have represented in the figure only one piston ; but it often happens that two or three are to be employed, and, in this case, the axis AE must carry four or six wipers, which should be so distributed upon its circumference, MECHANICS. 245 that when one piston is about to fall, the other may begin to rise. Now, in order that these pis- tons may be raised with an uniform motion, the form of the wiper mn must be the evolute of a circle, whose diameter is mm; or, in other words, it must be an epicycloid, formed by a generating circle, whose centre is infinitely distant, rolling upon the convex circumference of another circle, whose diameter is mm. But as a smafl roller P is frequently fixed to the extremity of the arm E to diminish the friction of the working parts, we must draw a curve within the above-mentioned involute, and parallel to it, the distance between them being equal to the radius of the roller ; l and this new curve will be the proper form for the wiper mn, when a roller is employed. The piston EF may also be raised or de- pressed uniformly, by giving a proper curvature to the arm PE, and fixing the roller upon the extremities of the bar mm. Thus, in Fig. 5, let CD be an axis, moved by any power, in which are fixed the arms DPI, MR, having rollers H, R, at their extremities, which act upon the curved arm op. When the piston EF is raised to the proper height, by the action of the roller H upon op, it then falls, and is again elevated by the arm M. In order that its motion may be uniform, the arm op must be part of a cycloid, the radius of whose generating circle is equal to the length of the arm DH, reckoning from its extremity H, or the centre of the roller, to the centre of the axle DC. But, when a roller is fix- ed upon the extremity H, we must draw a curve 1 The method of doing this is shewn, in Plate II, Fig. 4. See pages 218, 219. 246 MECHANICS. parallel to the cycloid, and without it, at the dis- tance of the roller's semidiameter; and this curve will be the proper form for the arm op. It is evident, that, when this mode of raising the pis- ton is adopted, the arm DH must be bent, as in the figure, otherwise the extremity p would pre- vent the roller H from acting upon the arm op. Plate iv, j n Yig. 6, we have another method of raising a weight perpendicularly with a uniform motion. Let AH be a wheel moved by any power which is sufficient to raise the weight MN by its extre- mity 0, from to e, in the same time that the wheel moves round one-fourth of its circum- ference, it is required to fix upon its rim a wing OBCDEH, which shall produce this effect with an uniform effort. Divide the quadrant OH into any number of equal parts Om, mn, &c. the more the better, and oe in into the same number ob, be, cd, &c. and through the points m, n, p, H, draw the indefinite lines AB, AC, AD, AE, and make AB equal to Ab, AC to Ac, AD to Ad, and AE to Ae ; then, through the points 0, B, C, D, E, draw the curve OBCDE, which is a portion of the spiral of Archimedes, and will be the proper form for the wiper, or wing, OHE. s It is evident, that when the point m has arrived at 0, the extre- mity of the weight will have arrived at b; because AB is equal to Ab; and, for the same reason, when the points n, p, H, have successively arrived at 0, the extremity of the weight will have arrived at the corresponding points c, d, e. The motion, there- fore, will be uniform; because the space described by the weight is proportional to the space describ- ed by the moving power, Ob being to Oc as Om 8 For a different way < ^forming this spiral, see Wolfii Opera Mathematics, torn, .', p. 3p. MECHANICS. 247 to On. If it be required to raise the weight MN with an accelerated or retarded motion, we have only to divide the line e, according to the law of acceleration or retardation, and divide the curve OBCDE as before. 2. When the lever moves upon a centre, the w ; c " ^ weight will rise in the arch of a circle, and con- S? in the sequently a new form must be given to the a . rcl j of wipers or wings. The celebrated Deparcieux, cir of the Academy of Sciences of Paris, has given an ingenious and simple method of tracing me- chanically the curves which are necessary for this purpose. Though this method was pub- lished about fifty years ago in the Memoirs of the Academy, it does not seem to be at all known to the mechanics of this country. We shall therefore lay it before the reader in as abridged and simplified a form as the nature of the subject will permit. Let A B, Fig. I, be a phteV lever lying horizontally, which it is required to lg ' x< raise uniformly through the arch B C into the position A C, by means of the wheel B F H, fur- nished with the wing B NO P, which acts upon the extremity C of the lever ; and let it be re- quired to raise it through B C in the same time that the wheel B FH moves through one half of its circumference ; that is, while the point M moves to B, in the direction MFB. Divide the chord CB into any number of equal parts, the more the better, in the points 1 , 2, 3, and draw the lines la 26 3c parallel to A B, or a hori- zontal line passing through the point B 3 and meeting the arch C B in the points a, b, c. Draw the lines CD, aD, ID, cD, and B D, cutting the circle BFH in the points m, ?i 9 o,p. Having drawn the diameter B M t divide the semicircle B FM into as many equal parts as the 248 MECHANICS. chord C B, in the points , &c. are not equal ; but the per- pendiculars let fall from the points c, a, b, &c. upon the horizontal lines, passing through a, b, &c. are "equal, being proportional to the equal lines cl; 1,2, Eucl, Vi, 2. Had it been required to raise the lever through equal arches, in- stead of equal heights, in equal times, then the arch BC, instead of its chord, would have been divided into equal parts. MECHANICS. 249 sg, us 9 Mu. The lever, therefore, has been raised uniformly, the ratio between the velocity of the power, and that of the weight, remaining always the same. If the wheel D turns in a contrary direction, according to the letters M.HB, we must divide the semicircle BHEM, into as many equal parts as the chord CB, viz. in the points e, g, i. Then, having set the arch B jn from e to d, the arch B n from g to f, and the rest in a similar manner, draw through the points d,f, h, E, the indefinite lines DR,DS,DT,DQ, make D R equal to D c ; D S equal to D b : DT equal to Da, and D Q equal to DC; and through the points B, R, S, T, Q, describe the spiral B R S TQ, which will be the proper form for the wing, when the wheel turns in the direction MHB. For, when the point e arrives at B 9 the point d will be in m, and R in c, where the extremity of the lever will now be found, hav- ing moved through Be in the same time that the power, or wheel, has moved through the division e B. In the same manner it may be shewn, that the lever will rise through the -equal heights cb, ba, aC, in the same time that the power moves through the corresponding spaces eg, gi, iM. The motion of the lever, therefore, and also that of the power, are always uniform. Of all the positions that can be given to the point B, the most disadvantageous are those which are nearest the points F, H ; and the most advanta- geous position is when the chord BC is vertical, passes, when prolonged, through jD, 'the 1 In the figure we have taken the point B in a disad- vantageous position, because the intersections are in this, case mobt distinct. 250 MECHANICS. centre of the circle. 2 In this particular case the two curves have equal bases, though they differ a little in point of curvature. The farther that the centre A is distant, the nearer do these curves resemble each other ; and if it were in- finitely distant, they would be exactly similar, and would be the spirals of Archimedes, as the extremity C would, in this case, rise perpendicu- larly. The intelligent reader will easily perceive, that 4, 6, or 8 wings may be placed upon the cir- cumference of the circle, and may be formed by dividing into the same number of equal parts as the chord BC, -J-, , ory, of the circumference, instead of the semicircle B FM. That the wing BNO may not act upon any part of the lever between A and C 9 the arm A C should be bent ; and that the friction may be di- minished as much as possible, a roller should be fixed upon its extremity C. When a roller is used, however, a curve must always be drawn parallel to the spiral described according to the preceding method, the distance between it and the spiral being everywhere equal to the radius of the roller. Mistake of When two or more wings are placed upon ^thb" 1 " tne circumference of the wheel, it has been the subject, custom of practical mechanics to make them por- tions of an ellipse whose semi-transverse axis is equal. to QZ), the greatest distance of the curve from the centre of the circle. But it will ap- pear, from a comparison of an elliptical arch with the spiral N, that it will not produce an uniform motion. If it should be required to raise the lever with an accelerated or retarded motion, we have only to divide the chord C, according to the degree of retardation or acce- leration required, and the circle into the same MECHANICS. 151 number of equal parts as before, and then de- scribe the curve by the method already illus- trated. As it is frequently more convenient to raise or depress weights by the extremity of a con- stant radius, furnished with a roller, instead of wings fixed upon the periphery of a wheel ; we shall now proceed to determine the curve which must be given to the arm of the lever, which is to be raised or depressed, in order that this ele- vation or depression may be effected with an uniform motion. Plate v, Let AE be a lever, which it is required to Fi s- * raise uniformly through the arch B C 9 into the position A C, by means of the arm or constant radius DE, moving upon D as a centre, in the same time that the extremity E describes the. arch EeF. From the point 6' draw C H at right angles to A B, and divide it into any number of equal parts, suppose three, in the points 1 , "2 ; and through the points 1, 2, draw la, 2&, paral- lel to the horizontal line AB, cutting the arch CJB'm the points , b, through which draw a A, bA. Upon D as a centre, with the distance Z), describe the arch EieF; and upon A as a centre, with the distance AD, describe the arch e D 9 cutting the arch EieF in the point e. Divide the arches E i , and d c from e to 0. Then through the points E, /, k, 0, and 0, t, /;, /'', draw the two curves ElhO, and OtpF, which will be 252 MECHANICS. the proper form that must be given to the arm of the lever. If the handle D E moves from E towards .F, the curve E must be used, but if in the contrary direction, we must employ the curve OF. It is evident, that when the extremity E of the handle D , has run through the arch E k, or rather /, the point / will be in , and the point z in #, because x z is equal to k /, and the lever will have the position Ab. For the same reason, when the extremity E of the handle has arrived at i, the point h will be in ?', and the point g in f, and the lever will be raised to the position A . Thus it appears, that the motion of the power and the weight are always propor- tional. When a roller is fixed at , a curve paral- lel to EO, or OF, must be drawn as formerly. It is upon these principles that the detent levers of clocks, and those connected with the striking part, should be formed. In every ma- chine, indeed, where weights are to be raised or depressed, either by variable or constant levers, its performance depends much on the proper form of the communicating parts. Fr : rm of 3 < Hitherto we have supposed, that the wheel wipers, , . , . . . , when they which carries the wipers, or wings, moves in the move in a same plane with the lever or weight to be raised, right^ngies Circumstances, however, often occur which ren- tothe lever d er ft necessary to elevate the lever by means of to ;e raised. i i J . , jl j a wheel moving at right angles to the plane in which the lever moves ; and when this method is adopted, a different form must be given to the wipers. As no writer on mechanics, so far as I know, has treated this subject, it becomes the more necessary to supply the defect by a few observations. MECHANICS. 253 Let ABC, Fig.' 3, be the lever which P. lateV > is to be raised round the axis AE, by the lg ' 3 action of the wing mn of the wheel D, upon the roller C, fixed at the extremity of the lever ; it is required to find the form which must - be given to the wiper mn. It is evident, from Fig. 4, where CB is a section of the lever and Fig. 4. roller, and EA the arch through which it is to be raised, that the breadth of the wiper must always be equal to mn, or rB, the versed sine of the arch BA, through which the roller moves, so that the extremity n of the wiper may act upon the roller B at the commencement of the motion, and that the other extremity m may act upon the roller A, when the lever arrives at the required position CA. It is easy to perceive, how- ever, that, if the acting surface mn of the wiper is always parallel to the horizon, or perpendicu- lar tothe radii of the wheel Z), or the plane in which it moves, it will act disadvantageously, except at the commencement of _lhe motion, when mn is parallel to CB. For, when mn has arrived at the position o/>, the extremity o will act upon the roller A, but in such an oblique and disadvantageous manner, that it will scarcely have any power to turn it upon its axis, or move the lever round the fulcrum C. The friction of the roller upon its axis, therefore, will increase, and the power of the wiper to turn the lever will diminish, in proportion to the length of the arch BA; and if CA arrives at a vertical position, the power of the wiper will be solely employed in wrenching the lever from its fulcrum. In order to avoid this inconvenience, we must endeavour to give such a form to the wiper, that its acting surface may always be parallel to the lever, or axis of the roller, having the 254 MECHANICS. position mn when the roller is at B, and the po- sition ob when the roller is at A. Having stated the peculiarities of this con- struction, let us now attend to the method by which the acting surface of the wiper must be formed. Since the lever CB is to be raised perpendicularly through the equal spaces re, ca, a A in equal times, the acting surface of the wiper must evidently be part of the spiral of Archimedes, 6 the method of describing which is shewn, in Fig. 6 of Plate IV ; but the difficulty lies in giving different degrees of in- clination to the acting surface, in order that the part in contact with the roller may be parallel Plate v, to the direction of the lever. Let AD, Fig. 6, be Fij.4 & 6. t j le W h ee l 9 w hich is to be furnished with wings, and let Cb the perpendicular height, through which the lever is to rise, be equal to Ar, in Fig. 4. Divide the quadrant Db into any number of equal parts, the more the better, suppose three, in the points c and r, and de- scribe the spiral of Archimedes DinC, as formerly directed. Divide Ar (Fig. 4) the sine of the arch BA, into the same number of equal parts, in the points c, a ; and draw of, eg parallel to CB, and cutting the circle in the points d, e, and the tangent Bb in the points/, g; and through the points C and d draw Chi. The line df is equal to the difference between radius and the cosine of the arch dB; fi is equal to the difference between the tangent and the sine of the same arch ; iB being the tangent, and fB equal to the sine of the arch dB, or angle dcB ; ad is equal to of- df, or to the difference be- See page 240. MECHANICS. 255 tween rf/and the versed sine of the whole arch^; fixda, and a k is equal to -^7- for an account of the similar triangles dft 9 dak, we have df:fi ~da:ak* 9 J ixd a and consequently ak T7~- Since then the points r, c, a 9 ^ (in Fig. 4) correspond respectively with the points Z), f, rc, 6' of the spiral, in Fig. 6, take/Y and set it from n to m, and A from n to ; take also g/i, and set it from i to A (Fig. 6) set cCand rA, in Fig. 6 and 4. The curves AnmC, and ON correspond with DkoC, in Fig. 6, and MP with DhmB. The diagonal curve MN corresponds with the diagonal curve DinC, and OM, the breadth of the wiper with mn, or rB, the versed sine of the arch AB, in Fig. 4-. The breadth OM, however, should always be a little greater than the versed sine of the arch through which the lever is to be raised, since MN is the path of the roller over the wiper's surface. Having thus described the different methods of raising weights, whether perpendicularly, or round a centre, with an uniform velocity and force, it would be unnecessary to apply the principles of construction to those machines which are formed for the elevation of weights. The practical mechanic can easily do this for himself. There is one case, however, which deserves pe- culiar attention, because the wipers, formed ac- cording to the preceding rules, will not produce the intended effect. This happens in the case of the large sledge hammer which is employed in plate v, forges. In Fig. 7, BC is the large hammer mov- F g- 7- ed round A as a centre, by means of the wiper wTTrsfor MW acting upon its extremity AC, or upon the raising roller R. The hammer must be tossed up with forge ham- a su( Jd en motion, so as to strike the elastic oaken spring E, which, being compressed, drives back the hammer, with great force, upon the anvil D. Now, if spiral wipers, constructed according to the directions already given, are employed, the hammer will indeed be raised equably without the least jolting, but it will rise no MECHANICS. 257 higher than the wiper lifts it, and will, therefore, fall merely with its own weight. But, if the wi- pers are constructed in the common way, and the hammer elevated with a motion greatly accelerat- ed, it will rise much higher than the wiper lifts it, it will impinge against the oaken beam E, and be repelled with great effect against the iron on the anvil D. In any of the preceding con- structions, this accelerated motion may be pro- duced, merely by dividing EC according to the Plate v, law of acceleration, and proceeding as already fis ' x & * directed. Vol. II. R MECHANICS. ON THE NATURE AND CONSTRUCTION OF WIND- MILLS. Description of a Wind-mill. x HE limited and imperfect manner in which Mr. Ferguson has treated of wind-mills, in the wind-mill, preceding volume, renders it necessary that the subject should now be prosecuted at greater length. The few observations which he has made, upon these machines, presuppose that the reader is acquainted with their nature and construction; a species of knowledge which is not to be ex- pected in the readers of a popular and elementary work. For the purpose of supplying this defect, and enabling the reader to understand the ob- servations, which may be made on the form and sition of the sails, and on the relative advan- es of horizonal and vertical wind-mills, we tan 1 1 give a description of a wind-mill, invented s^alMr. James Verrier, containing several im- bv ements on the common construction, for prov MECHANICS. 250 which the author was literally rewarded by the Society of Arts. This machine is represented in Fig. 1 , where p ^ te VI > AAA are the three principal posts, 27 feet 7j ' g< *' inches long, 22 inches broad at their lower ex- tremities, 18 inches at their upper ends, and 17 inches thick. The column B is 1 2 feet 2-^- inches long, 1 9 inches in diameter at its lower extremity, and 1 6 inches at its upper end ; it is fixed in the centre of the mill, passes through the first floor E 9 having its upper extremity secured by the bars GG. EEE are the girders of the first floor, one of which only is seen, being 8 feet 3 inches long, 1 1 inches broad, and 9 thick ; they are mortised into the posts AAA and the column , and are about 8 feet 3 inches distant from the ground floor. DDD are three posts 6 feet 4 inches long, 9 inches broad, and 6 inches thick ; they are mortised into the girders EF of the first and second floor, at the distance of 2 feet 4 inches from the posts A, &c. FFF are the girders of the second floor, feet long, 11 inches broad, and 9 thick ; they are mortised into the posts A y &c. and rest upon the upper extre- mities of the posts D, &c. The three bars GGG are 3 feet ly inches long, 7 inches broad, and 3 thick ; they are mortised into the posts D and the upper end of the column B, 4 feet 3 inches above the floor. P is one of the beams which support the extremities of the bray-trees, or brayers ; its length is 2 feet 4 inches, its breadth 8 inches, and its thickness 6 inches. / is one of the bray-trees, into which the extremity of one of the bridge-trees K is mortised. Each bray-tree is 4 feet 9^ inches long, g inches broad, and 7 thick ; and each bridge-tree is 4 feet 6 inches long, 9 inches broad, and 7 thick, R 2 26O MECHANICS. being furnished with a piece of brass on their upper surface to receive the under pivot of the millstones. LL are two iron screw-bolts, which raise or depress the extremities of the bray-trees. MMM are the three millstones, and NNN the iron spindles, or arbors, on which the turning millstones are fixed. is one of three wheels, or trundles, which are fixed on the upper ends of the spindles NNN; they are If) inches in diameter, and each is furnished with 14 staves, f is one of the carriage-rails, on which the upper pivot of the spindle turns, and is 4 feet 2 inches long, 7 inches broad, and 4 thick. It turns on an iron bolt at one end, and the other end slides in a bracket fixed to one of the joists, and forms a mortise, in which a wedge is driven to set the rail and trundle in or out of work ; Ms the horizontal spur-wheel that impels the trundles ; it is 5 feet 6 inches diameter, is fixed to the perpendicular shaft T, and is fur- nished with 42 teeth. The perpendicular shaft T is 9 feet 1 inch long, and 14 inches in dia- meter, having an iron spindle at each of its extremities ; the under spindle turns in a brass block fixed into the higher end of the column B ; and the upper spindle moves in a brass plate inserted into the lower surface of the carriage- rail C. The spur-wheel r is fixed on the upper end of the shaft T", and is turned by the crown- wheel v on the windshaft c, it is 3 feet 2 inches in diameter, and is furnished with 15 cogs. The carriage-rail C, which is fixed on the sliding kerb Z, is 17 feet 2 inches long, 1 foot broad, and 9 inches thick. YYQ is the fixed kerb, 37 feet 3 inches diameter, 14 inches broad, and JO thick, and is mortised into the posts MECHANICS. 261 and fastened with screw-bolts. The sliding kerb Z is of the same diameter and breadth as the fix- ed kerb, but its thickness is only 7-^- inches ; it revolves on 1 2 friction rollers fixed on the upper surface of the kerb YYQ, and has 4 iron half staples F, F, &c. fastened on its outer edge, whose perpendicular arms are 1O inches long, 2 inches broad, and 1 inch thick, and embrace the outer edge of the fixed kerb to prevent the sliding one from being blown off. The cap- sills JL, V, are 13 feet 9 inches long, 14 inches broad, and 1 foot thick ; they are fixed at each end, with strong iron screw bolts, to the sliding kerb, and to the carriage-rail C. On the right hand of w is seen the extremity of a cross rail, which is fixed into the capsills X and ^ by strong iron bolts : e is a bracket 5 feet long, id inches broad, and 10 inches thick; it is bushed with a strong brass collar, in which the inferior spindle of the windshaft turns, and is fixed to the cross rail w : b is another bracket 7 feet long, 4 feet broad, and 1O inches thick; it is fixed into the fore ends of the capsills, and, in order to embrace the collar of the windshaft, it is di- vided into two parts, which are fixed together with screw bolts. The windshaft c is 15 feet long, 2 feet in diameter at the fore end, and 18 inches at the other ; its pivot at the back end is 6 inches diameter, and the shaft is perforated to admit an iron rod to pass easily through it. The vertical crown wheel v is 6 feet in diameter, and is furnished with 54 cogs, which drive the spur wheel r. The bolster d, which is 6 feet 3 inches long, 1 3 inches broad, and half a foot thick, is fastened into the cross rail u>, directly under the centre of the windshaft, having a brass pulley fixed at its fore end. On the upper surface of R3 262 MECHANICS. this bolster is a groove, in which the sliding bolt R moves, having a brass stud at its fore end. This sliding bolt is not distinctly seen in the figure, but the round top of the brass stud is vi- sible below the letter h : the iron rod that passes through the windshaft bears against this brass stud. The sliding bolt is 4 feet 9 inches long, 9 inches broad, and y of a foot thick. At its fore end is fixed a line, which passes over the brass pulley in the bolster, and appears at a with a weight attached to its extremity, sufficient to make the sails face the wind that is strong enough for the number of stones employed ; and when the pressure of the wind is more than sufficient, the sails turn on an edge, and press back the sliding bolt, which prevents them from moving with too great velocity; and, as soon as the wind abates, the sails, by the weight , are pressed up to the wind, till its force is sufficient to give the mill a proper degree of velocity. By this appa- ratus, the wind is regulated and justly propor- tioned to the resistance or work to be performed ; an uniformity of motion is also obtained, and the mill is less liable to be destroyed by the rapidity of its motion. That the reader may understand how these effects are produced, we have represented, in plate vi, Fig. 2, the iron rod, and the arms which bear Fig. a. against the vanes ; ah is the iron rod which Method of p asses through the windshaft c, in Fig. 1 ; h is varying the r , . , . . ? . 7 , , angle of the the extremity, which moves in the brass stud that sails' inclin- i s fixed upon the sliding bolt; ai, at, &c. are the cross arms, at right angles to ah, whose ex- tremities z, z', similarly marked in Fig. 1, bear upon the edges of the vanes. The arms ai are 6f feet long, reckoning from the centre a, \ foot MECHANICS. 263 broad at the centre, and 5 inches thick 5 the arms n, n, &c. that carry the vanes or sails, are 1 8-j- feet long, their greatest breadth is 1 foot, and their thickness 9 inches, gradually diminish- ing to their extremities, where they are only 3 inches in diameter. The four cardinal sails, m, m, m, m 9 are each 1 3 feet long, 8 feet broad at their outer ends, and 3 feet at their lower ex- tremities ; p, /?, &c. are the four assistant sails, which have the same dimensions as the cardinal ones, to which they are joined by the line SSSS. The angle of the sail's inclination, when first op- posed to the wind, is 45 degrees, and regularly the same from end to end. It is evident, from the preceding description of this machine, that the windshaft c moves along with the sails; the vertical crown wheel v impels the spur wheel r, fixed upon the axis T, which carries also the spur wheel t. This wheel drives the three trundles //, one of which only is seen in the figure, which being fixed upon the spindles N, &c. communicate motion to the turning mill-stones. That the wind may act with the greatest Method of efficacy upon the sails, the windshaft or principal ^ D he axis must always have the same direction as wind. the wind. But as this direction is perpetually changing, some apparatus is necessary for bringing the windshaft and sails into their pro- per position. This is sometimes effected by supporting the machinery on a strong vertical axis, whose pivot moves in a brass socket firmly fixed into the ground, so that the whole ma- chine, by means of a lever, may be made to revolve upon this axis, and be properly adjusted to the direction of the wind. Most wind-mills, 264 MECHANICS. however, are furnished with a moveable roof, which revolves upon friction rollers inserted in the fixed kerb of the mill ; and the adjustment is effected by the assistance of a simple lever. As both these methods of adjusting the wind-shaft require the assistance of men, it would be very desirable that the same effect could be produced solely by the action of the wind. This may be done, by fixing a large wooden vane, or wea- ther-cock, at the extremity of a long horizontal arm, which lies in the same vertical plane with the windshaft. By this means, when the surface of the vane, and its distance from the centre of motion are sufficiently great, a very gentle breeze will exert a sufficient force upon the vane to turn the machinery, and will always bring the sails and windshaft to their proper posi- tion. This weathercock, it is evident, may be applied, either to machines which have a moveable roof, or which revolve upon a vertical arbor. wind-mills Prior to the French revolution, wind-mills wcre more numerous in Holland and the Netherlands than in any other part of the world, and there they seem to have been brought to a very high state of perfection. This is evident, not only from the experiments of Mr. Smeaton, from which it appears, that sails weathered in the Dutch manner produced nearly a maximum effect, but also from the observations of the celebrated Coulomb. This philosopher exa- mined above 50 wind-mills in the neighbour- hood of Lisle, and found that each of them performed nearly the same quantity of work when the wind moved with the velocity of ] 8 or 2O feet per second, though there were some MECHANICS. 265 trifling differences in the inclination of their windshafts, and in the disposition of their sails. From this fact, Coulomb justly concluded, that the parts of the machine must have been so disposed as to produce nearly a maximum ef-' feet. In the wind-mills, on which Coulomb's ex- Form of periments were made, the distance, from the ShtJmii extremity of each sail to the centre of the wind- according shaft, or principal axis, was 33 feet. The sails * were rectangular, and their width was a little more than 6 feet, 5 of which were formed with cloth stretched upon a frame, and the remaining foot consisted of a very light board. The line, which joined the board and the cloth, formed, on the side which faced the wind, an angle sensibly concave at the commencement of the sail, which diminished gradually till it vanished at its extremity. Though the surface of the cloth was curved, it may be regarded as com- posed of right lines perpendicular to the arm, or whip, which carries the frame, the extremities of these lines corresponding with the concave angle formed by the junction of the cloth and the board. Upon this supposition these right lines at the commencement of the sail, which was dis- tant about 6 feet from the centre of the wind- shaft, formed an angle of 60 with the axis, or windshaft, and the lines, at the extre- mity of the wing, formed an angle, increasing from 78 to 84, according as the inclina- tion of the axis of rotation to the horizon in- creased from 8 to 15 ; or, in other words, the greatest angle of weather was 30, and the least varied from 12 to 6, as the Inclination of the windshaft varied from 8 to 266 MECHANICS. 15V A pretty distinct idea of the surface of wind-mill sails may be conveyed, by conceiv- ing a number of triangles standing perpendicular to the horizon, in which the angle contained be- tween the hypothenuse and the base is constantly diminishing : the hypothenuse of each triangle will then be in the superficies of the vane, and they would form that superficies if their number were infinite. On the Form and Position of Wind-mill Sails. Form and M. Parent seems to have been the first mathe- poshionof matician who considered the subject of wind- mill sails in a scientific manner. The philoso- phers of his time entertained such erroneous opinions upon this point, as to suppose that the surface of the sails should be equally inclined to the direction of the wind and to the plane of their motion ; or, what is the same thing, that the angle of weather should be 45V But it appears, from the investigations of Parent, that a maximum effect will be produced when the sails are inclined 54|- to the axis of rota- tion, or when the angle of weather is Theincii. 35 j . In obtaining this conclusion, however, signed by M. P arent has assumed data which are inadmis- Parenter- sible, and has neglected several circumstances must materially affect the result of his in- 1 The weather of the sails is the angle which the surface of the sails forms with the plane of their motion, and is al- ways equal to the complement of the angle which that surface forms with the axis. 1 See Wolfii Opera Mathematica, torn, i, p. 680, where this angle is recommended. MECHANICS. 267 Vestigations. The angle, or inclination, assigned by Parent, is certainly the most efficacious for giving motion to the sails from a state of rest, 5 and for preventing them from stopping when in motion; but he has not considered that the action of the wind upon a sail at rest is different from its action upon a sail in motion : for since the extremities of the sails move with greater rapi- dity than the parts nearer the centre, the angle of weather should be greater towards the centre than at the extremity, and should vary with the velocity of each part of the sail. 4 The reason of this is very obvious. It has been demonstrated by Bossut,'" antf sufficiently established by ex- perience, that when any fluid acts upon a plain surface, the force of impulsion is always exerted most advantageously when the impelled surface is in a state of rest, and that this force diminishes as the velocity of the surface increases. J This may be demonstrated in the following manner. Let x be the cosine of the angle sought ; then, since the sine, cosine, and radius of any arch form a right angled triangle, the square of the sine will be'equal to the square of the co- sine subtracted from the square of the radius, that is, 1 l will be the square of the sine when the radius is unity. But the effect of the wind, on an oblique sail, is, in the com- pound ratio of the square of the sine of its obliquity, and the breadth of the sail projected on a plane perpendicular to the direction of the wind. Now, this breadth is exactly x, the cosine of the sail's inclination ; therefore x X 1 ** or x x' will represent the effectof the wind upon the sail. And, as this is to be a maximum, let us take its fluxion, which will be* 3x\x=0. Dividing by* wehave3jc*=:l,or xzry 7 -^ ' w hich is the cosine of 54" 44' 13". 4 See vol. i, p. g7> 5 Traite d'Hydrodynamique, 7/2. 'J68 MECHANICS. Now, let us suppose, with Parent, that the most advantageous angle of weather for the sails of wind-mills is 35y degrees for that part of the sail which is nearest the centre of rotation, and that the sail has everywhere this angle of \veather ; then, since the extremity of the sail moves with the greatest velocity, it will, in a manner, withdraw itself from the action of the wind ; or, to speak more properly, it will not receive the impulse of the wind so advantage- ously as those parts of the sail which have a less velocity. In order, therefore, to make up for this diminution of force, we must make the wind act more perpendicularly upon the sail, by diminishing its obliquity, that is, we must increase its inclination to the axis or the direction of the wind ; or, what is the same thing, we must diminish its angle of weather. But, since the velocity of every part of the sail is propor- tional to its distance from the centre of motion, every elementary portion of it must have a differ- ent angle of weather diminishing from the centre to the extremity of the sail. The law or rate of diminution, however, is still to be discovered, and we are fortunately in possession of a theorem of Euler's, afterwards given by M'Laurin, which theorem, determines this law of variation. 6 Let a represent the velocity of the wind, and c the velocity of any given part of the sail, then the effort of the wind upon that part of the sail will be greatest when the tangent of the angle of the wind's incidence, or of the sail's inclination to the axis, is to radius as v/2 + 9f_iJJ to 1. 2 a 6 Sec Maclaurin's Fluxions, art. 910 9 14. MECHANICS. 269 In order to apply this theorem, let us suppose that the radius or whip 7iis of the? ateVI sail a0$/, is divided into six equal parts, that' ... r l , , Explana- the point n is equidistant from m and j, and t ion and is the point of the sail which has the same a PP li f aion velocity as the wind; then, in the preced- ing theorem, we will have crra, when the sail is loaded to a maximum ; and therefore the tangent of the angle, which the surface of the sail at n makes with the axis, when a 1 will / 9 3 be v /2+--f--n3.561:r: tangent of 7T1Q', which gives 15 41' for the angle of weather at the point n. Since, at y of the radius c a, and since c is proportional to the distance of the corresponding part of the sail from the centre we will have, at y of the radius sm, czr , at ~ , t & of the radius, c=-j; at , c= ^> at i> c= y> and at the extremity of the radius, czr2a. By substituting these different values of c, instead of c in the theorem, and by making <7 1, the following table will be obtained, which exhibits the angles of inclination and weather which must be given to different parts of the sails. 270 MECHANICS. Table skewing the rate at which the inclination variet. Parts of tfie radius from the Velocity of the sail at these Angle made with the axis. Angle of weather. centre of distances or motion at s. values of c. Deg. Min. Deg. Min. ~6 a 3 63 26 26 34 2 1 la 69 54 20 6 for^ a 74 19 15 4 ** T 77 20 12 40 5 6 5a 79 27 10 33 I 2a 81 9 Having thus pointed out an important error in Parent's theory, and shewn how to find the law of variation in the angle of weather, we have farther to observe, that, in order to simplify the calculus, Parent supposed the velocity of the wind to be infinite when compared with the velocity of the sail, and that its impulsion upon the sail was in the compound ratio of the square of its velo- city and the square of the sine of incidence. The first of these suppositions is evidently inaccurate, and was shewn to be so by Daniel Bernouilli, in his Hydrodynamique. With regard to the force of impulsion on the sails, the proposition is per- fectly true in theory, and has been demonstrated MECHANICS. 271 by Pilot,' and other philosophers; but it un- questionably appears, from the experiments pre- sented to the French Academy, in 1763, by M. le Chevalier de Borda, and from those made, in 1776, by M. d'Alembert, the Marquis Condor- cet, and the Abbe Bossut, 2 that this proposition does not hold in practice. The first part of the Force of proposition, indeed, that the force of impulsion ^" is proportional to the square of the velocity of the surface*. surface that it is impelled, is true in practice ; but, when the angles of incidence are small, the latter part of the proposition must be abandoned, as it would afford very false results. In cases, how- ever, where the angles of incidence are between 50 and 9O degrees, we may regard the impulsion as proportional to the square of the velocity mul- tiplied by the square of the sine of incidence ; but we must remember, that the force thus de- termined by the theory will be a little less than that which would be found by experiment, and that the difference increases as the angle of inci- de^hce recedes from 90. Such boing the circumstances which Parent has overlooked in his investigations, we need not be surprised to find, from the experiments of Smeaton, that when the angle which he recom- mends was adopted, the sails produced a smaller effect than when they were weathered in the com- mon manner, or according to the Dutch con- struction. 3 1 Mem. de I'Acad. Paris, 1729, p. 540. * Nouvelles Experiences sur la resistance des fluides, par M. M. d'Alembert, le Marquis de Condorcet," et 1'Abbe Bossut, chap, v, 35. J Mons. Belidor has fallen into the same error as Parent, and observes, that the workmen at Paris make the angle of weather, 3 272 MECHANICS. iuier*sob- The theory of wind-mills has been treated at Unwind"* g reat l en g tn bv M * Euler, the most profound and mills. celebrated mathematician of his time. He has shewn, that the angle assigned by Parent is too small for a sail in motion, and that the angle of weather should vary with the velocity of the dif- ferent parts of the sails ; but, like Parent, he has supposed that the force of impulsion upon sur- faces, with different obliquities, is proportional to the square of the sines of their inclination. As the angles of incidence, however, are sufficiently great, this circumstance will have but a trifling effect upon his conclusions. After Euler has shewn, in general, how to determine the force of impulsion upon the sails, whatever be their figure and disposition, and whatever be the cele- rity of their motion ; he then investigates, by the method de maximis et minimis, what should be the inclination of the sails to the axis, and the velocity of their extremities, in order to produce a maximum effect ; and he finds, that this incli- nation and velocity are variable, and are inversely proportional to the momentum of friction in the machine. That the reader may fully understand this important result, we may remark, that, in theory, the greatest effect will be produced when the velocity of the sails is infinitely great, and when their surfaces are perpendicular to the wind's direction ; that is, when the angle of wea- ther is nothing. But both these suppositions are excluded in practice; for though the sails re- ceive the greatest possible impetus from the wind, weather 18% and thereby lose ^ of the effect; whereas, this is nearly the most efficacious angle that can be adopt- ed. See Architecture Hydra ulique> par Belidor, torn. 2, B. iii, pp. 3341. MECHANICS. 273 when they are inclined 90 to die axis, yet this force has not the smallest tendency to put them in motion; and it is not difficult to perceive, that the friction of the machine, and the resist- ance of the air to the thickness of the sails, must always limit the velocity of their motion. In this case, theory does not accord with practice ; but they may be easily reconciled, by making the angle of inclination 89 instead of 90, and supposing the sails to perform a finite, but a very great number of revolutions in a second, an hun- dred for example. Then the sails, having still a very disadvantageous position, will receive but a small impetus from the wind, which may be call- ed one pound. But this defect in the impelling power is made up by the great velocity of the sails ; and since the effect is always equal to the product of the weight and the velocity, we will have 1 X lOOlOO for the effect of the machine. Now, let us take friction into the account, and suppose it to be so great as to diminish the rapi- dity of the sails, from ; 100 to 5O turns in a second; then, in order that the machine may produce an effect equal to 100, as formerly, we must change the angle of the sail's inclination, till it receives from the wind an impetus equal to two pound for 2X5O 1OO. If the friction be still farther in- creased, the celerity of the machine will experi- ence a proportional diminution, and the angle of inclination must undergo such a change, that the force of impulsion received from the wind may make up for the velocity that is lost by an in- crease of friction. From these observations it plainly appears, that the celerity of the sails, and their inclination to the axis, depend upon the momentum of friction ; and as this is generally a constant quantity in machines, and can easily be Pol. II. S 274 MECHANICS, determined experimentally, the position of the sails, the velocity of their motion, and the effect of the machine, may be found from the following table, which is calculated from the formulae of Euler, and adapted to different degrees of fric- tion. In this table, F denotes the force of the wind upon all the sails ; d is the radius of the sail, or the distance of its extremity from the centre of the axis orwindshaft; v is the velocity of the wind ; and s the velocity of the sail's extremity, which is equal to the numbers contained in the fourth column. Table containing the Angle of Inclination and Weather of Wind- mill Sails, the Velocity of their Extremities^ and the Effect of the Machine % for any Degree of Friction. Angle of A np'lp Velocity of Effect of the Momentum th: sail'! AUglC the sails at Effect of the machine dif- of friction. inclination to the axis. of wea- ther. their extre- machine. ferently ex- mities. pressed. 0.235702 Fd 45 45 0.000000 v 0.000000 Fv O.OOOOOQF., 0.175837 Fd 50 40 0.127686 v 0.00471 8 Fv 0.036950 F s 0.1 2287 IFd 55 35 0.281334 v 0.017968 Fv 0.063869 Fs 0.079653 Fd 60 30 0.469882 v 0.037427 Fv 0.079653 F s 0.047001 Fd 65 25 0.711154v 0.0601 47 Fv 0.084576 Fs 0.024370 Fd 70 20 1.042160 v 0.083 159 Ft; 0.079795 Fs 0.01 0362 Fd 75 15 1.550395 v 0.1 03842 Fv 0.066978 F s 0.003084 Fd 80 10 2.499421 v 0.120105 Fv 0.048053 Fs 0-000386 Fd 85 5 5.208606 0. 1 30454 Ft/0.025046 F 4 0.000000 Fd 90 Infinite. 0.1 3400 IFu 0.000000 F s 1 2 3 4 5 6 P rece ding table has been applied, by table. Euler, solely to that species of wind-mills in which the sails are sectors of an ellipse, and which- intercept the whole cylinder of wind. This MECHANICS. 275 construction was recommended also by Parent ; but later and more accurate experiments have evinced, that when the whole area is filled up with sail, the wind does not produce its greatest effect, from the want of proper interstices to escape. On this account a small number of sails are generally used, and these are either rectan- gular, or a little enlarged at their extremities. It will be proper, therefore, to shew how the table can be applied to this description of sails, for the application is much more difficult than in the other case. It is evident, from the first column of the table, that before we can use it, we must find the value of F, or the force of the wind upon all the sails. But as this force depends not merely upon the quantity of surface, and the velocity of the wind, which are always given, but also upon the angle of their inclination, which is unknown, some method of determining it, independently of this angle, must be adopted. Euler has shewn how to do this, in the case where the whole area is filled with elliptical sectors ; but there is no direct method of determining the value of F in the case of rectangular sails, when the angle of inclination is unknown. We must find it there- fore by approximation ; that is, we must take any probable angle of inclination, 7O for example, and find the value of F suited to this angle, and thence the co-efficient of Fd, in the first column. With this co-efficient enter the table, and take out the corresponding angle of inclination, which will be either less or greater than 7O. With this new angle of inclination find a more accurate value of F, and consequently a new co-efficient of F d. If this co-efficient does not differ very much from that formerly found, S 2 276 MECHANICS. it may be regarded as true, and employed for taking out of the table a more accurate angle of inclination, along with the velocity of the sails, and the effect of the machine. We shall now illustrate both these methods by an example, after having shewn how to determine by experi- ment the momentum of friction, and the velo- city of the wind. Tojind the Momentum of Friction. On the ma- Jn a calm day, when the wind-mill is unload- frSio f edj or performing no work, bring two opposite sails into a horizontal position; and, having attached different weights to the extremities of their radius, find how many pounds are sufficient not only for impressing the smallest motion on the sails, but for continuing them in that state ; and the number of pounds multi- plied into the length of the radius, will be the momentum of friction. When this experiment is made, it will always be found that a greater weight is necessary for moving the sails than for continuing them in motion ; and, in order that the quantity of friction may be accurately esti- mated, the wind-mill should be put in motion immediately before the experiment is made; for the friction always increases with the time in which the communicating parts have remained in contact. Tojind the Velocity of the Wind. Various instruments, denominated anemome- ters, or anemoscopes, have been invented for MECHANICS. 277 measuring the force and velocity of the wind, the best of which are those which were con- structed by Mr. Pickering ' and Dr. Lind. * The velocity of the wind has been deduced also from the motion of the clouds, and the change effect- ed by the wind upon the motion of sound. 3 The second of these methods is manifestly inac- curate, and the first takes for granted what is palpably erroneous, that the velocity of the wind is the same in the higher regions of the atmo- sphere, as at the surface of the earth. The in- genious Professor Leslie having found, in the course of his experiments on heat, that the re- frigerant, or cooling, power of a current of air is exactly proportional to its velocity, derives, from this principle the construction of a new and 1 Philosophical Transactions, No. 473. * Id. vol. Ixv, p. 353. J Brisson, Traitedt Physique, vol. ii, p. 150, 1015. Foi* the description of another anemometer, see Wolfii Opera Math. torn, i, p. 773. The anemometer invented by Bou guer is described in his Traite du Navlre, p. 35Q ; Onsen Bray's anemometer in the Mem. Acad. Paris, 1734; and another by Zeiher, which is a combination of Bouguer's instrument, with the apparatus employed by Smeaton, is described in the Nov. Com. Petrop. 1766, yoL x, p. 302. I have seen an ingenious anemometer, invented by the Rev. Mr. Jamefon of St. Mungo, and founded on the same principle as the quadrant, described by Bossut, for finding the velocity of running water. A plane surface, suspended in a vertical direction, is exposed to the action of the wind, and the angle of its elevation, to the tangent of which the force of the wind is always proportional, is pointed out by an index carried round by a wheel and pinion. By means of a fmall click which falls into the teeth of one of the wheels, the plain surface, or pendiflum, is detained in the position to which it is raised, and the greatest velocity of the wind may be determined in the absence of the observer. S 3 anemome- ter. 278 MECHANICS. Leslie's simple anemometer. e It is in reality nothing more,' says he, ' than a thermometer, only with its bulb larger than usual. Holding it in the open still air, the temperature is marked : it is then warmed by the application of the hand, and the time is noted which it takes to sink back to the middle point. This I shall term the fundamental measure of cooling. The same observation is made on exposing the bulb to the impresson of the wind, and I shall call the time required for the bisection of the inter- val of temperatures, the occasional measure of cooling. After these preliminaries, we have the following easy rule : Divide the funda- mental by the occasional measure of cooling, and the excess of the quotient above unit, being multiplied by 4-f, will, express the velocity of the wind in miles per hour. The bulb of the thermometer ought to be more than half an inch in diameter, and may, for the sake of portability, be filled with alcohol tinged, as usual, with archil. To simplify the observation, a sliding scale of equal parts may be applied to the tube. When the bulb has acquired the due temperature, the zero of the slide is set opposite to the limit of the coloured liquor in the stem ; and, after having been heated, it again stands at 20 in its de'scent, the time which it thence takes until it sinks to 10 is measured by a stop watch. Extemporaneous calculation may be avoided, by having a table engraved upon the scale for the series of occa- sional intervals of cooling.' 4 4 Enquiry into the Nature and Propagation of Heat;, p. 284. MECHANICS. 279 The most simple method of determining the Coulomb's velocity of the wind, is that which Coulomb em- ployed in his experiments on wind-mills, and which requires neither the aid of instruments nor the trouble of calculation. 5 Two persons were placed on a small elevation, at the distance of 150 feet from one another, in the direction of the wind ; and, while the one observed, the other measured the time which a small and light feather employed in moving through the space. The distance between the two persons, divided by the number of seconds, gave the velocity of the wind per second. Having thus shewn how to find the momentum of friction, and the velo- city of the wind, we shall now explain the use of the table. Supposing the radius of the sails to be 20 feet, the velocity of the wind 1O feet per second, and[!" c f t] that it requires a force of 10 pounds acting at the ' extremity of the radius to overcome the friction of the machine, it is required to find the angle of weather, the velocity of the sails, and the ef- fect of the machine. Let d, the radius of the sails, be 20 feet, then the momentum of friction will be 1OX2O rr20O pounds. Let n, the number of sails, be 1 2, while a represents the breadth of the sails at their extremities, and b the breadth into which they are projected, or the breadth which they would occupy if reduced into a plane perpendicu- lar to the wind. Then, since the whole cylinder of wind is supposed to be intercepted, the effect produced upon all the oblique sails will be equal See Mem. de PAcad. Paris, 1781, p. 70. 28O MECHANICS, to the effect that would be produced upon a per- pendicular surface, equal to the whole area of the polygon into which the oblique triangular sails are projected. The value of b 9 there- fore, may be found by plane trigonometry, the length of the sail and the angle of the polygon being given, or by the following the- 18O orem : b zz 2d x tang. , d being radius, and n n the number of sails. In the present case 18O* then, we shall have b 'm 2 X 20 x tang. , n or r=40Xtang.!5, = 10.717968 feet. Now, since the area of any triangle is equal to its alti- tude multiplied by half its base, the area of a polygon will be equal to the altitude of one of the triangles which compose it, or to the radius pf the inscribed circle, multiplied by half the number of its sides. The area of the polygon, therefore, into which the sails are projected, or the quantity of perpendicular surfece impelled by the wind, will be |- ndb, and, consequently, the force of impulsion F 9 upon this surface, will be j- ndbvv, where vv is tfre square of the wind's velocity, to which the force of impulsion is al- ways proportional. In the present case, then, the force .F, which impels the sails, will be 6X2O X 1O.71 7968xw; and if w be the altitude which is due to the velocity of the wind, or the height through which a heavy body must fall in order to acquire that velocity, the force -t>f im- pulsion F will be equal to the weight of a mass of air, whose volume is 1286.15616XW cubic feet, or to 1^- vv cubic feet of water ; for water is about 80O times more dense than air ; that is to 100 vv pounds avoirdupois, 62^ of which are * * MECHANICS. 281 equal to a cubic foot of water. But, in order that the machine may move, the momentum of friction 2OO must be less than O.235702XjFW, or 0.235702X100^X20; for when it is exactly this, the wind cannot move the machine, as appears from the first line of the table; or, what is the same thing, the height due to the velocity of the wind, viz. vv must be greater than 0.424, or 4 of a foot, which corresponds to a velocity of 5.222, or 5|. Unless, there- fore, the celerity of the wind exceeds 5~ feet per second, it will not be able to move the ma- chine. These things being premised, let us now proceed to determine the construction and effect of the machine, upon the supposition that the momentum/ of friction is 20O pounds, and the velocity of the wind 10 feet per se- cond. Now, vu, the height due to this velo-. city is 1-f- feet; 1 therefore the force of impul- sion F is = 1OO w pounds, or 10OXJ-, or = 16O pounds avoirdupois; and Fd = 1(X) X 2O= 3200. But the momentum of friction, viz. Fd 9 multiplied into its co-efficent, should be equal 20O pounds; therefore, the co-efficient 20O 20O will be equal to 7w" = ^5o :=0t0 ^ 2500 ' ^ ^ e momentum of friction will be O.0625OO Fd. With this number enter the first column of the table, and you will find the angle of inclina- tion corresponding to it to be about 63 ; the * The height answering to any velocity, and the velo- city due to any height, may be found by the following theorems, in which v is the velocity, and h the height due toil; v=20 feet in an hour, Mem. del'Acad. Berlin, 1783, p. 333. A very interesting discussion on the force of men, by Lambert, will be found in the Mem. de 1'Acad. Berlin, 1/76. MECHANICS. 283 viding this quantity by 80O, we will have 127 F=-^-vv cubic feet of water, and muldply- 4OO - ing this by 62^ we will have F 19.8 w pounds avoirdupois. Now, let the velocity of the wind be 30 feet per second, the height vv due to this velocity will be 14 feet nearly; and, conse- quently F= 19.8X1 4, = 276 pounds avoirdupois. Fd will therefore be = 554O; and, since the whole momentum of friction is 20O, the co-ef- 20O ficient of Fd will be =77^=0.036101, and the OOMJ momentum of friction, expressed as the table re- quires, will be = O.O36101 Fd. Having entered the table with this number, the proper angle of inclination will be found to be 67-f degrees. With this angle, instead of 70, repeat the fore- going calculation, and after finding a new co-efficient to Fd, enter the table with it a second time, and you will have the proper angle of in- clination, differing but little from the former, and likewise the velocity of the sails, and the ef- fect of the machine. 3 By comparing with the preceding theory the performance of the wind-mills examined by Cou- lomb and Lulofs, 1 it will be found that their power is almost double of that which is deduced from theory. This remarkable difference arises * Those who wish to inquire farther into the theory of wind-mills, will find some excellent observations in D'Alembert's Traite de 1'Equilibre et du Mouvement des Fluides, 177> P 396, 368; or in his Opatcules, torn, v, p. 148, &c. and also by Lambert, in the Mem. de J'Acad. Berlin, 1775, p. 92. 8 See pages 287, 288. 284 MECHANICS. from a defect in the common hypothesis, which represents the force of impulsion as proportional to the square of the wind's velocity, and the square of the sine of the angle ot incidence. When the wind impinges upon the sail, the air behind it is rarefied ; this rarefaction increases with the velocity of the wind, and therefore the impulsion must be much greater than what is deduced from the common hypothesis. Euler supposes it to be twice as great ; and, upon this supposition, has treated the subject more accur- ately in a subsequent memoir, 8 which, however, is too profound to be of any service to the prac- tical mechanic. Results of These theoretical deductions, however inter- smeaton's estm g they may be, must yield in point of prac- expen- . o J J ' *; _ ments. tical utility to the observations or our country- man Mr. Smeaton. From a variety of well con- ducted experiments, he found, that the com- mon practice of inclining plane sails, from 72 to 75, to the axis, was much more efficacious than the angle assigned by Parent, the effect being as 45 to 31. When the sails were wea- thered in the Dutch manner, that is, when their surfaces were concave to the wind, and when the angle of inclination increased towards their ex- . tremities, they produced a greater effect than when they were weathered either in the com- mon way, or according to Maclaurin's theo- rem. 4 But when the sails were enlarged at their extremities, as represented at a/S, so that a was one third of the radius m /, and a m to 5 Recherches plus exactes sur 1'effet des moulins a vent, Mem. de 1'Acad. Berlin, J766> vol. xii, p. 164. , 4 See page 262. MECHANICS. 28J m , as 5 to 3, their power was greatest of aU, though the surface acted upon by the wind re- mained the same. 5 If the sails be farther en- larged, the effect is not increased in proportion to the surface ; and, besides, when the quantity of cloth is great, the machine is much exposed to injury by sudden squalls of wind. In these experiments of Smeaton, the angle of weather varied with the distance from the axis ; and he found, from several trials, that the most effica- cious angles were those contained in the follow- ing table : Parts of the radius Angle with the Angle of weather. tut, which is divided axis. 1 into 6 parts. 1 72 18 2 71 19 3 72 18 middle 4 74 16 5 77^ 121 6 83 7 Supposing the radius ms of the sail, to be BO feet, then the sail will commence at -g- ms, or 5 feet from the axis, where the angle of inclination will be 72. At ms, or 10 feet from the axis, the angle will be 71, and so on, 5 In the sails used in Portugal, the broad part is placed at the end of the arm. They are much more swolu than those of common wind-mills, and may be set to draw, in a manner smilar to the stay sails of a ship. 286 MECHANICS. On the Effect of Wind-mill Sails. Effect of The following maxims, deduced by Mr. Smea- wmd-mill r , . -11. sails, ton from his experiments, contain the best in- formation which we have upon the effect of wind- mill sails, if we except a few experiments made by Coulomb. According Maxim 1. The velocity of wind-mill sails, whe- to Smea- , 111111 i ten . ther unloaded or loaded, so as to produce a maxi- mum effect, is nearly as the velocity of the wind, their shape and position being the same. Maxim 2. The load at the maximum is near- ly, but somewhat less than, as the square of the velocity of the wind, the shape and position of the sails being the same. Maxim 3. The effects of the same sails at a maximum, are nearly, but somewhat less than, as the cubes of the velocity of the wind. Maxim 4. The load of the same sails, at the maximum, is nearly as the squares, and their ef- fect as the cubes of their number of turns in a given time. Maxim 5. When sails are loaded, so as to pro- duce a maximum at a given velocity, and the ve- locity of the wind increases, the load continuing the same ; lt, The increase of effect, when the increase of the velocity of the wind is small, will be nearly as the squares of those velocities ; 2 dl y, When the velocity of the wind is double, the effects will be nearly as 1O to 27^- ; but, 3 dl y, When the velocities compared are more than double of that where the given load produces a maximum, the effects increase nearly in the simple ratio of velocity of the wind. MECHANICS. 287 Maxim 6. In sails where the figure and po- sition are similar, and the velocity of the wind 'the same, the number of turns, in a given time, will be reciprocally as the radius or length of the sail. Maxim 7. The load, at a maximum, which sails of a similar figure and position will overcome, at a given distance from the centre of motion, will be as the cube of the radius. Maxim 8. The effects of sails of similar figure and position are as the square of the radius. Maxim 9. The velocity of the extremities of Dutch sails, as well as of the enlarged sails, in all their usual positions when unloaded, or even loaded to a maximum, are considerably quicker than the velocity of the wind. 6 M. Coulomb made a number of experiments According on wind-mills that were employed to raise stamp- ers for the purpose of bruising seed. He found that wind-mills having the dimensions formerly stated, 7 produced an effect equivalent to 100O pounds raised through the space of 218 feet in a minute. The quantity of force, which was lost by the action of the wipers upon the stampers, was equal to 100O pounds raised through 16^ feet in a minute ; and the friction was equivalent to 10OO pounds raised through 18-^ feet in a mi- 6 Mr. Smeaton found, when the radius was 30 feet, that for every three turns of the Dutch sails in their common position, (when the angle of weather at the extremity is nothing), the wind-mill moves at the rate of two miles an hour ; for every five turns in a minute of the Dutch sails, in their best position, the wind moves four miles an hour ; and for every six turns in a minute of the enlarged sails, in their best position, the wind will move five miles an hour. 7 See page 266. 288 MECHANICS. nute. The total quantity of action, therefore, exerted by the wind in moving the machine, was equal to 1000 pounds elevated to the height of 253 feet in a minute, the velocity of the wind being 20 feet per second. It appears, too, from Coulomb's experiments, that when the wind moved at the rate of 13 feet per second, the sails made 8 turns in a minute ; when the velocity of the wind was 20 feet per second, the sails performed 13 turns in a minute; and when its velocity was 28 feet in a second, the sails made 1 7 turns in a minute. 8 By taking the medium of these results, it will be found, that the number of turns made by the sails in a mi- nute, is to the number of feet which the wind moves in a second, as 1 to 1.6. Hence, when the velocity of the sails is given, that of the wind may be easily determined. M. Lulofs of Leyden examined a Dutch wind- mill, which was employed to drain marshes, and found that when the wind moved at the rate of 30 feet per second, it was capable of raising 1500 cubic feet of water, 4 feet high, in a minute. The wind-mill had four rectangular sails, each being 43 feet long, and 95|- feet broad ; and the mean angle of weather was 1 7 degrees. On Horizontal Wind-mills. Horizontal A variety of opinions have been entertained re- wind-miiis. S p ec tj n g the relative advantages of horizontal and vertical wind-mills. Mr. Smeaton, with great justice, gives a decided preference to the latter ; but, when he asserts that horizontal wind-mills 8 Mem. de 1'Acad, Paris, '1781, p. 81. MECHANICS. 289 have only ~ or -^_ of the power of vertical ones, he certainly forms too low an estimate of their power. Mr. Beatson, on the contrary, who has received a patent for the construction of a new horizontal wind-mill, seems to be prejudiced in their favour, and greatly exaggerates their com- parative value. From an impartial investigation, it will probably appear, that the truth lies be- tween these two opposite opinions; but before entering on this discussion, we must first consi- der the nature and form of horizontal wind- mills. In Fig. 3 of Plate VI, CK is the perpendicular Plate vi, axis, or windshaft, which moves upon pivots. Four ^ 2 ' cross bars CA, CD, IB, FG, are fixed to this arbor, which carry the frames APJB, DEFG. The sails A I, EG, are stretched upon these frames, and are carried round the axis CK, by the perpendicular impulse of the wind. Upon the axis CK a toothed wheel is fixed, which gives morion to the particular machinery that is employed. In the figure only two sails are re- presented ; but there are always other two placed at right angles to these. Now, let the Common sails be exposed to the wind, and it will bef iethodot i i r bringing evident that no motion will ensue; for the back the force of the wind, upon the sail A I, is coun- 8 ? 118 *?*? 13 ' . . . r &C Wind. teracted by an equal and opposite force, upon the sail EG. In order, then, that the wind may communicate motion to the machine, the force upon the returning sail EG must either be removed by screening it from the wind, or diminished by making it present a less sur- face when returning against the wind. The first of these methods is adopted in Tartary, and in some provinces of Spain ; but is ob- l. II. T 29O MECHANICS. jected to by Mr. Beatson, from the inconveni- ence and expence of the machinery and attend- ance requisite for turning the screens into their proper positions. Notwithstanding this objec- tion, however, I am disposed to think that this is the best method of diminishing the action of the wind upon the returning sails, for the moveable screen may easily be made to follow the direction of the wind, and assume its proper position, by means of a large wooden weather- cock, without the aid either of men or machi- nery. It is true, indeed, that the resistance opposed to the returning sails is not completely removed ; but it is at least as much diminished as it can be by any method hitherto proposed. Besides, when this plan is resorted to, there is no occasion for any moveable flaps and hinges, which must add greatly to the expence of every other method. The mode of bringing the sails back against method. fa e ^n^ which Mr. Beatson invented, is, per- haps, the simplest and best of the kind. He makes each sail AI to consist of six or eight flaps, or vanes, AP b 1, b I c 2, &c. moving upon hinges, represented by the dark lines AP, b I, c 2, &c. so that the lower side b 1, of the first flap overlaps the hinge, or highest side, of the second flap, and so on. When the wind, therefore, acts upon the sail AI, each flap will press upon the hinge of the one immediately below it, and the whole surface of the sail will be exposed to its action. But when the sail AI returns against the wind, the flaps will re- volve upon their hinges, and present only their edges to the wind, as is represented at EG 9 so that the resistance occasioned by the return of MECHANICS. 29JI the sail must be greatly diminished, and the motion will be continued, by the superiority of force exerted upon the sails, in the position A I. In computing the force of the wind up- Resistance on the sail A I, and the resistance opposed tqjj^**" if by the edges of the flaps in EG, Mr. Beatson sails, finds, that, when the pressure upon the former is 1872 pounds, the resistance opposed by the latter is only about 36 pounds, or T ' T part of the whole force ; but he neglects the action of the wind upon the arms CA, &c. and the frames which carry the sails, because they expose the same surface, in the position AI 9 as in the posi- tion EG. This omission, however, has a tend- ency to mislead us in the present case, as we shall now see, for we ought to compare the whole force exerted upon the arms, as well as the sail, with the whole resistance which these arms and the edges of the flaps oppose to the mo- tion of the wind-mill. By inspecting Fig. 3, it F g-3. will appear, that if the force, upon the edges of the flaps, which Mr. Beatson supposed to be 12 in number, amounts to 36 pounds, the force spent upon the bars CD, DG 9 GF 9 FE 9 &c. cannot be less than 60 pounds. Now, since these bars are acted upon with an equal force, when the sails have the position AI 9 1872-f-SO = 1932 will be the force exerted upon the sail A I, and its appendages, while the opposite force, upon the bars and edges of the flaps, when re- turning against the wind will be 36+60=96 pounds, which is nearly ^ of 1Q32, instead of -^ as computed by Mr. Beatson. Hence we may see the advantages which will pro- bably arise from using a screen for the re- Burning sail, instead of moveable flaps, as it will 292 MECHANICS. preserve not only the sails, but the arms and the frame which support it, from the action of the wind. 6 Campari- We shall now conclude this article with a few son be- remarks on the comparative power of horizon- ticai and tal and vertical wind-mills. It was already stat- e( *' t ^ lat ^ r ' Smeaton rather under-rated the ' former, while he maintained that they have only -| or ~ the power of the latter. He observes, that when the vanes of a horizontal and vertical mill are of the same dimensions, the power of the latter is four times that of the former ; because, in the first case, only one sail is act- ed upon at once ; while, in the second case, all the four receive the impulse of the wind. This, however, is not strictly true, since the vertical sails are all oblique to the direction of the wind. Let us suppose that the area of each sail is 10O square feet ; then the power of a horizontal sail will be 100, as only one sail is acted upon, and as its surface is perpendicular to the wind, and the power of a vertical sail may be call- , . 2 ed 10O X sine 70= 88 nearly, (7O being the common angle of inclination ;) but since there are four vertical sails, the power of them all will be 4X88=352; so that the power of the horizontal sail is to that of the four vertical ones, as 1. to 3.52, and not as 1 to 4 accord- 6 The sails of horizontal wind-mills are sometimes fixed, like float-boards, on the circumference of a large drum or cylinder. These sails move upon hinges so as to stand at right angles to the drum, when they are to re- ceive the impulse of the wind ; and when they return against it, they fold down upon its circumference. See Repertory of Arts, vol. vi. MECHANICS. 293 ing to Mr. Smeaton. But Mr. Smeaton also observes, that if we consider the farther dis- advantage which arises from the difficulty of get- ting the sails back against the wind, we need not wonder if horizontal wind-mills have only about i or the power of the common sort. We have already seen, that the resistance occa- sioned by the return of the sails, amounts to -~ of the whole force which they receive ; by sub- tracting ^, therefore, from ^ we will find that the power of horizontal wind-mills is only j, or little more than 7 less than that of ver- tical ones. This calculation proceeds upon a supposition, that the whole force exerted upon vertical sails is employed in turning them round the axis of motion ; whereas, a considerable part of this force is lost in pressing the pivot of the axis, or windshaft, against its gudgeon. Mr. Smeaton has overlooked this circumstance, otherwise he could never have maintained that the power of four vertical sails was quadruple the power of one horizontal sail, the dimensions of each being the same. Taking this circum- stance into the account, we cannot be far wrong in saying, that, in theory at least, if not in prac- tice, the power of a horizontal wind-mill, is about y or 7 of the power of a vertical one, when the quantity of surface and the form of the sails is the same, and when every part of the ho- rizontal sails have the same distance from the axis of motion as the corresponding parts of the vertical sails. But if the horizontal sails have the position A I, EG, instead of the po- Fig. 3. sition CAdm, CD on, their power will be greatly increased, though the quantity of surface is the same ; because the part CP 3m being T3 294 MECHANICS. transferred to BI 3 d, has much more power to turn the sails. I would recommend it also to the mechanic, to furnish horizontal wind-mills with six or eight sails ; for, as it happens in the ana- logous case of water-mills, the wind bends round their extrenlties, and impinges upon those parts of the sail immediately behind, which are not ex- pofed to the direct action of the wind. Having these methods, therefore, of increasing the power of horizontal sails, we would encourage every attempt to improve their construction, as not only laudable in itself, but calculated to be of es- sential utility in a commercial country. MECHANICS. JVj.R. FERGUSON, in his fourth lecture, treated the subject of wheel-carriages with great ria 6 c *- perspicuity, and has communicated much prac- tical information of considerable importance. Many of the prejudices, however, which he has there encountered, and several others which have escaped his notice, still continue to prevail in this country ; and as some of these have been countenanced even by ingenious men, we are laid under a more urgent necessity of at- tempting to develope the source of their errors, and of regulating the practice of the mechanic by the deductions of theory. The very assistance which theory has, in this case, furnished to the artist, has been rendered not only useless, but in- jurious, by an erroneous application ; and we may safely affirm, that there is no species of machin- ery where less science is displayed than in the construction and position of carriage-wheels. The few imperfect hints, which we are able to convey upon this subject, regard the formation and position of the wheels, the line of traction* 296 MECHANICS. and the method of disposing the load which is to be drawn. To some of these we solicit the reader's attention, as being entirely new, and apparently leading to consequences of high im- portance. On the Formation of Carnage Wheels. Wheels act When the wheels of carriages either move as mecha- U p On a level surface, or overcome obstacles powers, which impede their progress, they act as mecha- nical powers, and may be reduced to levers of the first kind. In order to elucidate this remark, which is of great importance in the present dis- piateii cussion, let^ be the centre, and BCN the cir- Fig. 5. ' cumference of a wheel 6 feet in diameter, and A PP- let the impelling power P, which is attached to the extremity of a rope ADP 9 passing over the pulley Z), act in the horizontal direction AD. Then, if the wheel is not affected by friction, it will be put in motion upon the level surface MJB 9 when the power P is infinitely small. For since the whole weight of the wheel rests on the ground at the point J3, which is the fulcrum of the lever AB 9 the distance of the weight from the centre of motion will be nothing, and there- fore the mechanical energy of the smallest power P, acting at the point A, with a length of lever Af$, will be infinitely great when compared with the resistance of the weight to be raised ; and this will be the case however small be the lever AB^ and however great be the weight of the wheel. But as the wheels of carriages are constantly meeting with impediments, let C be an obstacle 6 inches high, which the wheel is to surmount. Then the spoke AC will represent the lever, C MECHANICS. 297 its fulcrum, AD the direction of the power ; and if the wheel weighs 1OO pounds, we may repre- sent it by a weight IV fixed to the wheel's centre A, or to the extremity of the lever CA, and acting in the perpendicular direction AB, in opposition to the power P. Now, the mechani- cal energy of the weight IV to pull the lever round its fulcrum in the direction AE, is repre- sented by CE, while the mechanical energy of an equal weight P to pull it in the opposite di- rection AF, is represented by CF; an equilibri- um, therefore, will be produced, if the power P is to the weight IV as CE to CF, or as the sine is to the cosine of an angle, whose versed sine is equal to the height of the obstacle to be surmounted ; for EB, the height of the mound C, is the versed sine of the angle BAG, and CE is the sine, and CF the cosine of the same angle. In the present case, where EB is 6 inches, and AB 3 feet, EB, the versed sine, will be 1666, &c. when AB is 1000 ; and, consequently, the angle BAG will be 33 33', and CE will be to EF as 52 to 83, or as 66 to 100. A weight P 9 therefore, of 66 pounds, acting in a horizontal direction, will balance a wheel 6 feet diameter, and I OO pounds in weight, upon an obstacle 6 inches high ; and a small additional power will enable it to surmount that obstacle. But if the direction AD of the power be inclined to the ho- rizon, so that the point D may rise towards H y the line FC, which represents the mechanical energy of P, will gradually increase, till DA has reached the position HA, perpendicular to AC, where its mechanical energy, which is now a maximum, is represented by AC the ra- dius of the wheel ; and since EC is to CA as 53 to IOO, a little more than 53 pounds will be 298 MECHANICS. sufficient for enabling the wheel to overcome the obstacle. Proceeding in this way, it will be found, that the power of wheels to surmount emi- nences increases with their diameter, and is di- rectly proportional to it, when their weight re- mains the same, and when the direction of the power is perpendicular to the lever which acts against the obstacle. Hence we see the great advantages which are to be derived from large wheels, and the disadvantages which attend small Advan- ones. There are some circumstances, however, bfge f which confine us within certain limits in the use wheels, of large wheels. When the radius AB of the wheel is greater than DM the height of the pulley, or of that part of the horse to which the rope or pole DA is attached, the direction of the power, or the line of traction, AD will be oblique to the horizon, as Ad, and the mechani- cal energy of the power will be only Ae, whereas it was represented by AE when the line of trac- tion was in the horizontal line DA. Whenever the radius of the wheel, therefore, exceeds four feet and a half^ the height of that part of the horse, to which the traces should be attached, 1 the line of traction AD will incline to the hori- zon, and, by declining from the perpendicular AH, its mechanical effort will be diminished ; and, since the load rests upon an inclined plane, the trams, or poles, of the cart will rub against the flanks of the horse, even in level roads, and 1 According to M. Couplet, the distance of this part of the horse from the ground is generally three feet and a half, (Mem. dc 1'Acad. Paris, 1733, 8 VO , p. 75). In horses of a common size, however, it is seldom below four feet and a half. MECHANICS. 299 still more severely in descending ground. Not- withstanding this diminution of force, however, arising from the unavoidable obliquity of the im- pelling power, wheels exceeding four and a half feet radius have still the advantage of smaller ones ; but their power to overcome resistances does Hot increase so fast as before. Hitherto we have supposed the weight of the large and small wheels to be the samej but it is evident, that, when we augment their dia- meter, we add greatly to their weight ; and, by thus increasing the load, we sensibly diminish their power. From these remarks, we see the superiority of great wheels to small ones, and the particular circumstances which suggest the propriety of making the wheels of carriages less than 4^ feet radius. Even this size is too great, as we shall afterwards shew, when speaking of the line of traction ; and we may safely assert, that they ought never to exceed 6 feet in diameter, and should never be less than 87- feet. When the nature of the machine will permit, large wheels should always be preferred, and small ones should never be adopted, unless we are compelled to employ them by some unavoidable circumstances in the construction.* This maxim, which has * For the advantage of those who wish to study this subject with greater attention, and with the view also of recommending the use of large wheels, we shall subjoin the following references to the works of eminent men, who have held the same opinion upon this point ; Mersennus' Geom. p. 459; Herigon, Mecan. prob. xvi, Schol.j Wallis' Mccan. c. vii, prob. 3, Schol. 15 ; Phil. Trans, vol. xv, p. 856 j Camus' Traite des Forces Mouvantes, prop, xxviii, xxx ; and Deparcieux sur IcTirage des Chevaux, Mem. de I'Acad. Paris, 1760, p. 263, 4 to . SOO MECHANICS* been inculcated by every person who has written on the subject, seems to have been strangely ne- glected by the practical mechanics of this coun- The fore- try. The fore-wheels of our carriages are still wheels of una ccountably small, and we have seen carts carnages . ' . ' . , . too small, moving upon wheels scarcely jourteen inches in diameter. The workman, indeed, will tell us, that, in the one case, the wheels are made small for the conveniency of turning, and, in the other, for facilitating the loading of the cart ; but how trifling are these advantages when com- pared with that diminution of the horses* power, which necessarily results from the use of small wheels. A convenient place for turning with large fore wheels, which is not frequently re- quired, may be procured, by going to the end of a street ; and a few additional turns of a windlass will be sufficient to raise the heaviest load into carts which are mounted upon high wheels. It has been objected against large fore- wheels, that the horses, when going down a de- clivity, cannot so easily prevent the carriages from running downwards ; but this very objec- tion, trifling as it is, is a plain confession that large fore-wheels are advantageous, both in hori- zontal and inclined planes, otherwise their tend- ency downwards would not be greater than that of small ones. 3 From some experiments on wheel carriages, Mr. Walker conceives that the greatest advantage was obtained when the hind wheels were 5 feet 6 inches in diameter, and the fore ones 4 feet 8 inches, whereas the large wheels are, in general, only 4 feet 8 inches, and the small ones 3 feet 8 inches. System of Familiar Philosophy, vol. i, p. 130. MECHANICS. 301 Having thus ascertained the superiority ofonthe large wheels, we are now to determine on the^JJ^* 1 shape which ought to be assigned them. Every person, who is not influenced by preconceived notions, would affirm, without hesitation, that, if the wheels are to consist of solid wood, they should be portions of a cylinder; and if they Cylindrical are to be composed of naves, spokes, and fellies, w that the rim of the wheel ought to be cylin- drical, and the spokes perpendicular to the naves. But some men, desirous of being in- ventors, have renounced this simple shape, and adopted the more complicated form of Fig. 6, where the rim EsrA is conical, and the PIate n , . ,. , , -r,, ., , Fie. o. Apii. spokes inclined to the naves. 4 Philosophers, too, have found a reason for this change, and it wheels. has been adopted in every country, more from the authority of names than the force of argu- ment. 5 It is with the greatest diffidence, how- ever, that we presume to contradict a practice which has been defended by the most celebrated mechanics ; but we trust that the reader's indul- gence will be proportioned to the solidity of the reasons upon which this difference of sentiment is founded. ^ Advanta- The form represented in Fig. 6, then, isgesanddh- liable to two objections, namely, the inclination * f "ncsnre of the spokes, and the conical figure of the dishing wheels. 4 This inclination is about 1 inch out of 11, or A is generally 3 inches when the diameter ot the wheel is 5 ' feet. ! I have seen some carriage wheels, in which one half of the spokes were inclined about one fourth more than the other half, every alternate spoke being equally inclined to the axis. The reasons for such a construction I have not been able to discover. 302 MECHANICS. rim. When the spokes are inclined to the nave, the wheels are said to be concave, or dishing, and they are recommended by Mr. Ferguson, and every other writer on mechanics, from the numerous advantages which are said to attend them. By extending the base of the carriage, they prevent it from being easily overturned, they hinder the fellies from rubbing against the load or the sides of the cart, and when one wheel falls into a rut ; and, therefore, supports more than one half of the load, the spokes are brought into a perpendicular position, which ren- ders them more capable of supporting this addi- tional weight. Now, it is evident, that the se- cond of these advantages is very trifling, and may be obtained when it is wanted, by interposing a piece of board between the wheel and the load. The other two advantages exist only in very bad roads ; and if they are necessary, which we very much question, in a country like this, where the roads are so excellently made, and so regularly repaired, they can easily be procured by making the axle-tree a few inches longer, and increasing the strength of the spokes. But it is allowed, on all hands, that perpendicular spokes inclination are preferable on level ground. The inclination spoke^ dis- f tne spokes, therefore, which renders concave advantage- wheels advantageous in rugged and unequal roads, ge ~ ren d ers them disadvantageous when the roads are in good order ; and where the good roads are more numerous than the bad ones, as they certainly are in this country, the disadvantages of concave wheels must overbalance their ad- vantages. It is true, indeed, that in concave wheels, the spokes are in their strongest position when they are exposed to the severest strains ; that is, when one wheel is in a deep rut, and MECHANICS. 303 sustains more than one half of the load ; but it is equally true, that in level ground, where the spokes are in their weakest position, a less severe strain, by continuing for a much longer time, may be equally, if not more, detrimental to the wheel. s Upon these observations, we might rest the Concave opinion which we have been maintaining, and tSf appeal for its truth to the judgment of every more inju- intelligent and unbiassed mind ; but we shall JJJJJ^ the go a step farther, and endeavour to shew, that concave dishing wheels are more expensive, more injurious to the roads, more liable to be broken by accidents, and less durable, in gene- ral, than those wheels in which the spokes are perpendicular to the naves. By inspecting Fig. ^kte ii 6, it will appear, that the whole of the pres- *' sure, which the wheel AB sustains, is exert- ed along the inclined spoke ps 9 and therefore acts obliquely upon the level ground nD, whe- ther the rims be conical or cylindrical. This oblique action must necessarily injure the roads, by loosening the stones more between B and D than between B and n ; and if the load were sufficiently great, the stones would start up between s and D. The texture of the roads, indeed, is sufficiently firm to prevent this from * Mr. Anstice, in his excellent treatise on wheel car- riages, recommends concave wheels ; but candidly allows, that ' some disadvantages attend this contrivance ; for the carriage thus takes up more room upon the road, which makes it more unmanageable ; and when it moves upon plain ground, the spokes not only do not bear perpen- dicularly, by which means their strength is lessened, but the friction upon the nave and axle is made unequal, jjnd the more so the more they are dished.' 312 MECHANICS. taking place ; but, in consequence of the ob- lique pressure, the stones between v and D will, at least, be loosened, and, by admitting the rain, the whole of the road will be materially damag- ed. But when the spokes are perpendicular to the nave, as /m, and when the rims mA, nB are cylindrical, or parallel to the ground, the weight sustained by the wheel will act perpen- dicularly upon the road, and however much that weight is increased, its action can have no tendency to derange the materials of which it is composed ; but is rather calculated to consoli- date them, and render the road more firm and durable. And more It was observed, that concave wheels are more expensive than plain ones. This addition- al expence arises from the greater quantity of wood and workmanship which the former re- quire ; for, in order that dishing wheels may be of the same perpendicular height as plane ones, the spokes of the former must exceed in length those of the latter, as much as the hypothenuse oA of the triangle oAm exceeds the side om ; Plate ir, and therefore the weight and the resistance of Fi e- 6 - such wheels must be proportionally great. The inclined spokes, too, cannot be formed nor in- serted with such facility as perpendicular ones. The extremity of the spoke which is fixed into the nave is inserted at right angles to it, in the direc- tion op, and if the rims are cylindrical, the other extremity of the spoke should be inserted in a si- milar manner, while the intermediate portion has an inclined position. There are, therefore, two flexures or bendings in the spokes of concave wheels, which require them to be formed out of a larger piece of wood than if they had no such MECHANICS. 305 flexures, and render them liable to be broken by any sudden strain at the points of flexure. In the comparison which we have now been stating, between the merits of concave and plane wheels, we have taken for granted what has been uniformly stated by the advocates of the former, that when one of the wheels falls into a rut or surmounts an eminence, the lowest sustains much more than one half of the load. Now, though it be true that the lowest wheel supports more than one half of the load, yet we deny that it bears so much as has generally been supposed, 6 and we shall prove the assertion, by pointing out a method of ascertaining the additional weight which is transferred to one wheel by any given elevation of the other. Let p kte n, AMOC represent a cart loaded with coals or Flg ' 7 ' lime, or any other material which fills it to the^ top, and let AB be a horizontal line on the sur- borne by face of a level road. Then, if the wheel A re- mains fixed, and the wheel C raised to any of them ; height, its lower extremity C will describe the ? umount '. i -ns\ i i_ s i M i -ing an emi arch BO round the centre A, while the centre oinence. gravity D of the whole machine and load will move in the arch JVM round the same centre. Now, let us suppose that BC is an eminence which the wheel C has to surmount, and that it has arrived at the top of it ; it is required to find what proportion of the load is sustained by each wheel. Bisect the horizontal line AB in e, and from e draw ed at right angles to AB 9 and meet- 6 Mr. Ferguson observes, (vol. i, p. 116), that the wheel, which falls into the rut, bears much more of the weight than the other ; and, a little afterwards, that it bears most ofthe weight of the load. n. u 306 MECHANICS. ing the arch NM in the point d, join AC, A d, AD, and from the point D let fall the per- pendicular DE. The point d will be the centre of gravity of the load when the points C and B coincide ; that is, when the wheels are resting on the horizontal plane AB. For since, in this case-, each wheel bears an equal part of the weight, the line of direction, or a vertical line passing through the centre of gravity, will cut the base AE, so that Ae will be to eB as the weight upon the wheel A to the weight upon C- r and, therefore, ed will be the line of direction, and the point d, where it cuts the circle NM, in which the centre of gravity moves, will be the centre of gravity of the load in a horizontal po- sition. Now as D is the centre of gravity when the cart is in its inclined position, the perpendi- cular DE will be the line of direction, and the weight sustained by the wheel A will be to that sustained by C as EB to EA, or Ee will repre- sent the additional weight transferred upon A, when AB represents the whole of the load. But Ee can be easily determined for any value of BC, the height of the obstacle. For, while the point C moves from B to C, the centre of gra- vity rises from d to D, so that Dd and BC are similar arches, and AB, Ad, BC, are known, AB being the distance between the wheels, and Ad being equal to the square root of the sum of the squares of Ae, the half of that distance, and de the height of the centre of gravity (Eucl. 1 , 47), and BC being the height of the eminence. But since de, the sine of the arch dN 9 is known, dN is known, and also DN, the sum of the two arches, Dd, dN. The cosines AE, Ae, of the arches DN, dN, are therefore known, and con- sequently Ee, their difference may be determin- MECHANICS* ed ; or, otherwise, Ee is the difference of the versed sines EN, eN, of the same arches. Let us now take a particular value of BC, or rather of Co, the perpendicular height of the eminence, and call it 1 2 inches ; for even in the worst roads there are few eminences which are greater than this. Let AB, the distance between the wheels, be 6 feet, and de, the height of the centre of gravity, 4 feet, then Co will be of the radius AS, 'and making //5=:1OOOOOO, CO will be 166666, which, being, the natural sine of the arch BC, gives 9 35' for the arch BC, and for the similar arch Dd. Now, since Ae is 3 feet, and de 4 feet, the sum of their squares will be 25, and its square root 5 will be the length of the hypothenuse Ad, or the radius of the circle NDM. Then, making Ad radius, or 10OOOOO, de, the sine of the arch dN, will be \ of it, or 80OOOO; and therefore the arch dN will be 53 8', and the arch DN, 62 43'. But AE, the cosine of the arch DN, is =r 458S91 r: or _iA. 5 nearly, of AD 5 feet, and is therefore equal to 2 feet 3 inches and 6 tenths ; conse- quently EeAe AE will be 8 inches and 4 tenths, which is nearly f of AB. We may there- fore conclude, that the additional weight sustain- ed by the wheel A, while the other wheel is ris- ing over an obstacle 12 inches in perpendicular height, is j only of the whole load ; or that -| of the pressure upon the wheel C is transferred to the wheel A, while surmounting an eminence 1 2 inches high. If one of the wheels faHs mro a rut 12 inches deep, the same conclusion wfll result ; and we may affirm, that as the ruts and eminences which are generally to be met with even in bad roads, are for the most part much less than 1 2 inches in depth or height, such a U 2 SOS MECHANICS. small proportion of the load will be transferred to the lowest wheel, that there is no necessity for inclining the spokes in order to sustain the additional weight. When the cart is loaded with stones, or any heavy substance, the centre of gra- vity will be lower than d, so that a less propor- tion of the weight will be transferred to one wheel by the elevation of the other ; and when it is loaded with hay, or any light material, the lowest wheel will sustain a greater proportion of the load. Concave w e s hall now dismiss the subject of concave ty injured!* wheels with one observation more, and we beg the reader's attention to it, because it appears to be decisive of the question. The obstacles which carriages have to encounter, are almost never spherical protuberances, which permit the ele- vated wheel to resume by degrees its horizontal position. They are generally of such a nature, that the wheel is instantaneously precipitated from their top to the level ground. Now the momentum with which the wheel strikes the ground is very great, arising from a successive accumulation of force. The velocity of the wheel C is considerable when it reaches the top of the eminence, and while it is tumbling into the hori- Fig. 7. zontal line AE^ the centre of gravity is falling through the arch Dd, and the wheel C is receiv- ing gradually that proportion of the load which was transferred to A, till, having recovered the whole, it impinges against the ground with great velocity and force. But in concave wheels, the spoke which then strikes the ground is in its weakest position, and therefore much more lia- ble to be broken by the impetus of the fall, than the spokes of the lowest whefel by the mere trans- ference of additional weight. Whereas if the MECHANICS. 3O9 spokes be perpendicular to the nave, they re- ceive this sudden shock in their strongest posi- tion, and are in no danger of giving way to the strain. In the preceding observations, we have sup- PLATE posed the rims of the wheels to be cylindrical, Fl - 6 - as AC, BD. In concave wheels, however, the rims are uniformly made of a conical form, as AT^ Bs> which not only increases the disadvan- tages that we have ascribed to them, but adds many more to the number. Mr. Gumming, in Conicaj a late treatise on wheel carriages, solely devot- ed to the consideration of this single point, has shewn, with great ability, the disadvantages of conical rims, and the propriety of making them cylindrical ; but we are of opinion that he has ascribed to conical rims several disadvantages which arise chiefly from an inclination of the spokes. He insists much upon the injury done to the roads by the use of conical rims ; yet, though we are convinced that they are more in- jurious to pavements and highways than cylindri- cal rims, we are equally convinced, that this in- jury is occasioned chiefly by the oblique pressure of the inclined spokes. The defects of conical rims are so numerous and palpable, that it is wonderful how they should have been so long overlooked. Every cone that is put in motion upon a plane surface, will revolve round its ver- tex, and if force is employed to confine it to a straight line, the smaller parts of the cone will be dragged along the ground, and the friction greatly increased. Now when a cart moves up- on conical wheels, one part of the cone rolls while the other is dragged along, and though confined to a rectilineal direction by external force, their natural tendency to revolve round IT 3 "31O MtCHANICii. their vertex occasions a great and continued fric- tion upon the linch pin, the shoulder of the axle- tree, and the sides of deep ruts. The shape of the wheels being thus determin- ed, we must now attend to some particular parts of their construction. The iron plates of which the rims are composed should never be less than 3 inches in breadth, as narrower rims sink deep into the ground, and therefore injure the roads and fatigue the horses. Mr. Walker, indeed, at- tempts to throw ridicule upon the act of parlia- ment which enjoined the use of broad wheels, but he does not assign any sufficient reason for his opinion, and ought to have known, that se- veral excellent and well devised experiments were lately instituted by Boulard and Margijeron, 7 which evinced, in the most satisfactory manner, the great utility of broad wheels. Upon this subject an observation occurs to us which has not been generally attended to, and which ap r pears to remove all the objections which can be urged against broad rims. When any load is supported upon two points, each point supports, one half of the weight ; if the points are increas- ed to four, each will sustain one fourth of the load, and so on, the pressure upon each point of support diminishing as the number of points increases. If a weight, therefore, is supported by a broad surface, the points of support are in- finite in number, and each of them will bear an infinitely small portion of the load ; and, in the same way, every finite portion of this surface 7 The memoir which contains an account pf these ex* periments, was presented to the Academy of Lyons, and i published in the Journal de Physique, torn. ^Jx, p. 424. TV J -.-;- :. -j -# t fT"T; ."** ** ' 1 MECHANICS 311 sustain a part of the weight inversely pro- portional to the number of similar portions which the surface contains. Let us now suppose that a cart, carrying a load of 1 6 hundred weight, is supported upon wheels whose rims are 4 inches in breadth, and that one of the wheels passes over 4 stones, each of them an inch broad, and equally high, and capable of being pulverized only by a pressure of 40O weight. Then, as each wheel sustains one half of the load, and as the wheel which passes over the stones has 4 points of support, each stone will bear a weight .of 20O weight, and therefore will not be broken. But if the same cart, witfc rims only 2 inches in, breadth, should pass the same way, it will cover only 2 of the stones ; and the wheel having now only two points of support, each stone will be pressed with a weight of 40O weight, and will therefore be reduced to powder. Hence we may infer, that narrow wheels are, in another point of view, injurious to the roads, by pulverizing the materials of which they are composed* As the rims of wheels wear soonest at their Practical edges, they should be made thinner in the mid- remarks - die, and ought to be fastened to the fellies with nails of such a kind, that their heads may not rise above the surface of the rim. In some mi- litary waggons, we have seen the heads of these nails rising an inch above the rims, which not only destroys the pavements of streets, but op- poses a continual resistance to the motion of the wheel. If these nails were 8 in number, the wheel would experience the same resistance as if jt had to surmount 8 obstacles, 1 inch high, dur- ing every revolution. The fellies on which the rims are fixed should, in carriages, be 87 inches -deep, and in waggons 4 inches. The nayes 312 MECHANICS. should be thickest at the place where the spokes are inserted, and the holes in which the spokes are placed should not be bored quite through, as the grease upon the axle-tree would insinuate it- self between the spoke and the nave, and pre- vent that close adhesion which is necessary to the strength of the wheel. On the Position of the position of it mus t naturally occur to every person reflect. s ' ing upon this subject, that the axle-trees should be straight, and the wheels perfectly parallel, so that they may not be wider at their highest than at their lowest point, whether they are of a coni- cal or a cylindrical form. In this country, how- ever, the wheels are always made concave, and *ke en( k ^ ^ e ax ^ e - trees are universally bent e- downwards, in order to make them spread at the V ecs> top and approach nearer below. In some carriages which we have examined, where the wheels were only 4 feet 6 inches in diameter, the distance of the wheels at top was fully 6 feet, and their dis- tance below only 4 feet 8 inches. By this fool- ish practice, the very advantages which may be derived from the concavity of the wheels are completely taken away, while many of the dis- advantages remain ; more room is taken up in the coach-house, and the carriage is more liable to be overturned, by the contraction of its base. With some mechanics it is a practice to bend the ends of the axle-trees forwards, and thus make the wheels wider behind than before. This blunder has been strenuously defended by Mr. Henry Beighton, who maintains that wheels in this position are more favourable for turning, MECHANICS. 313 since, when the wheels are parallel, the outer- most would press against the linch-pin, and the innermost would rub against the shoulder of the axle-tree. In rectilineal motions, however, these converging wheels engender a great deal of fric- tion, both on the axle and the ground, and must therefore be more disadvantageous than parallel ones. This, indeed, is allowed by Mr. Beigh- ton ; but he seems to found his opinion upon this principle, that as the roads are seldom straight lines, the wheels should be more adapted for curvilineal than for rectilineal motion. In what part of the world Mr. Beighton has examined the roads we cannot say ; but of this we are sure, that there are no such flexures in the roads of Scotland. On the Line of Traction, and the Method by which Horses exert their Strength M. Camus, a gentleman of Lorrain, was the Line of first person who treated on the line of traction. 8 tractlon ' He attempted to shew that it should be a hori- zontal line, or rather that it should always be parallel to the ground on which the carriage is moving, both because the horse can exert his greatest strength in this direction, and because the line of draught being perpendicular to the vertical spoke of the wheel, acts with the largest possible lever. M. Couplet, 9 however, consi- dering that the roads are never perfectly level, 8 Traite des Forces Mouvantes, p. 387. 9 Reflexions, sur le tirage des charrettes, Mem. dc I'Acad. Paris, 1733, 8vo PP' 75> 8 ^ 314 MECHANICS. / and that the wheels are constantly surmounting small eminences, even in the best roads, recom r mends the line of traction to be oblique to the FIAT* ii, horizon. By this means the line of draught HA y (which is by far too much inclined in the figure), will in general be perpendicular to the lever AC which mounts the eminence, and will therefore act with the longest lever when there is the great- est necessity for it. We ought to consider also, that when a horse pulls hard against any load, he always brings his breast nearer the ground, and therefore it follows, that if a horizontal line of traction is preferable to all others, the direction of the traces should be inclined to the horizon when the horse is at rest, in order that it may be Horizontal when he iowers his breast and exerts his utmost force. How horses The particular manner, however, in which Jiv* ' m g agents exert their strength against great loads, seems to have been unknown both to Camus and Couplet, and to many succeeding writers upon this subject. It is to M Deparcieux, an exceL lent philosopher and ingenious mechanic, that we are indebted for the only accurate informa- tion with which we are furnished ; and we are sorry to see, that philosophers who flourished a ter him have overlooked his important instruc- tions. In his Memoir on the draught of horses, 1 hejias shewn, in the most satisfactory manner, that animals draw by their weight, and not by the force of their muscles. In four-footed ani- mals, the hinder feet is the fulcrum of the lever by which their weight acts against the load, and 1 Sur le Tirage des Chevau?, published in the Mem. de 1'Acad. Paris, 1760, 4'% p. 263, 8 VO , p. 27^. MECHANICS* when the animal pulls hard, it depresses its chest, and thus increases the lever of its weight, and diminishes the lever by which the load resists its efforts. Thus, let P be the load, DA the PLATE n, line of traction, and let us suppose FC tobe Fi s-'' the binder leg of the horse, AY part of its body, A its chest or centre of gravity, and CE the level road. Then AFC will represent the crooked lever by which the horse acts, which is equivalent to the straight one AC. But when the horse's weight acts downwards at A,* round C as a centre, so as to drag forward the rope AD, and raise the load P, CE will represent the power of the lever in this position, or the le- ver of the horse's weight, and CF the lever by which it is resisted by the load, or the lever of resistance. Now, if the horse lowers its centre of gravity A, which it always does when it pulls hard, it is evident that CE, the lever of its weight, will be increased, while CF, the lever of its re- sistance, will be diminished, for the line of trac- tion AD will approach nearer to CE Hence we see the great benefit which may be derived from large horses, for the lever AC necessarily increases with their size, and their power is al- ways proportioned to the length of this lever, their weight remaining the same. Large horses, therefore, and other animals, will draw more than small ones, even though they have less muscular /orce, and are unable to carry such a heavy bur- den. The force of the muscles tends only to 1 It may be imagined that the fore feet of the horse pre- vent it from acting in this manner ; but Deparcieux has fihewo by experiment that the fore feet bear a much less part pf the hone's weight when he draws than when he is a; rest. 516 MECHANICS. make the horse carry continually forward his cen- tre of gravity ; or, in other words, the weight of the animal produces the draught, and the play and force of its muscles serve to continue it. 3 position of From these remarks, then, we may deduce the trition. proper position of the line of traction. When the line of traction is horizontal, as AD, the le- ver of resistance is CF; but if this line is oblique to the horizon, as Ad, the lever of resistance is diminished to Cf, while the lever of the horse's weight remains the same. Hence it appears, that inclined traces are much more advantageous than horizontal ones, as they uniformly diminish the resistance to be overcome. Deparcieux, how- ever, has investigated experimentally the most favourable angle of inclination, and found, that when the angle DAF, made by the trace Ad, and a horizontal line is 14 or 15 degrees, the horses pulled with the greatest facility and force. This value of the angle of draught will require the height of the spring-tree bar, to which the traces are attached in four-wheeled carriages, to be one half of the height of that part of the horse's breast to which the fore end of the traces is con- nected. 4 Notwithstanding the great utility of inclined 3 When I first compared Deparcieux's theory with the manner in which horses appear to exert their strength, I was inclined to suspect its accuracy ; but a circumstance occurred which removed every doubt from my mind. I ob- served a horse making continual efforts to raise a heavy load over an eminence. After many fruitless attempts, it raised its fore feet completely from the ground, pressed down its head and chest, and instantly surmounted the obstacle. 4 This height is about 4 feet 6 inches, and therefore the height of the spring-tree bar should be only 2 feet 3 inches, whereas it is generally 3 feet. MECHANICS. 317 traces, it will not be easy to derive complete ad- vantage from them in two-wheeled carriages with- out diminishing the size of the wheels. In all four-wheeled carriages, however, they may be easily employed ; and in many other cases where wheels are not concerned, great advantage may be derived from the discovery of Deparcieux. On tJte Position of the Centre of Gravity, and the manner of Disposing the Load. From Mr. Ferguson's observations on the cen- Position of tre of gravity, 5 it must be evident, that if the Jj e "vi" axle-tree of a two-wheeled carriage passes through the centre of gravity of the load, the carriage will be in equilibrio in every position in which it can be placed with respect to the axle-tree, and in going up and down hill, the whole load will be sustained by the wheels, and will have no ten- dency either to press the horse to the ground or to raise him from it. But if the centre of gravi- ty is far above the axletree, as it must necessa- rily be according to the present construction of wheel carriages, a great part of the load will be thrown on the back of the horses from the wheels, when going down a steep road, and thus tend to accelerate the motion of the carriage, which the animal is striving to prevent ; while in ascending steep roads a part of the load will be thrown behind the wheels, and tend to raise the horse from the ground, when there is the greatest necessity for some weight on his back, to enable him to fix his feet on the earth, and Vol. i, page 17. 518 MECHANICS. overcome the great resistance which is occasion* ed by the steepness of the road. On the contra- ry, if the centre of gravity is below the axle, the horse will be pressed to the ground in going up hill, and lifted from it when going down. In all these cases, therefore, where the centre of gra- vity is either in the axietree, or directly above or below it, the horse will bear no part of the load in level ground : In some situations the ani- mal will be lifted from the ground when there is the greatest necessity for his being pressed to it, and he will sometimes bear a great proportion of the load when he should rather be relieved from it. The only way of remedying these evils is to assign such a position to the centre of gravity $ that the horse may bear some portion of the load when he must exert great force against it ; that is, in level ground, and when he is as- cending steep roads ; for no animal can pull with its greatest effort, unless it is pressed to the ground. Now, this may, in some measure, plate 11, be effected in the following manner. Let *** BCN, be the wheel of a cart, AD one of the shafts, D that part of it where the cart is sus- pended on the back of the horse, and A the axietree ; then, if the centre of gravity of the load is placed at m, a point equidistant from the two wheels, but below the line DA, and before the axietree, the horse will bear a certain weight on level ground, a greater weight when he is going up hill and has more occasion for it, and a less weight when he is going down hill and does not require to be pressed to the ground. All this will be evident from the figure, when we recollect, that if the shaft DA is horizontal, the centre of gravity will press more upon the MECHANICS. 319 point of suspension D the nearer it cdfoes to it ; or, the pressure upon D, or the horse's back, will be proportional to the distance of the centre of gravity from A. If m, therefore, be the centre of gravity, bA will represent its pressure upon ), when the shaft DA is horizontal. When the cart is ascending a steep road, AH will be the position of the shaft, the centre of gravity will be raised to , and a A will be "the pressure upon D. But if the cart is going down hill, AC will be the position of the shaft, the centre of gravity will be depressed to n, and cA will represent the pressure upon the horse's back. The weight sustained by the horse, therefore^ is properly regulated, by placing the centre of gravity at m. We have still, however, to deter- mine the proper length of ba and ;, the dis- tance of the centre of gravity from the axle, and from the horizontal line DA ; but these de- pend upon the nature and inclination of the roads, upon the length of the shaft DA, which varies with the size of the horse, on the magni- tude of the load, and on other variable circum- stances, it would be impossible to fix their value. If the load along with the cart weighs 4OO pounds, if the distance DA be 8 feet, and if the horse should bear 50 pounds of the weight, then I' A ought to be 1 foot, which being -5- of DA, will make the pressure upon D exactly 5O pounds. If the road slopes 4 inches in 1 foot, bm must be 4 inches, or the angle bAm should be equal to the inclination of the road, for then the point m will rise to a when ascending such a road, and will press, with its greatest force, on the back of the horse. When carts are not constructed in this man- Method c ner, we may, in some degree, obtain the 320 MECHANICS. end, by judiciously disposing the load. Let us suppose that the centre of gravity is at 0, when the cart is loaded with homogeneous materials* such as sand, lime, &c. then, if the load is to consist of heterogeneous substances, or bodies of different weights, we should place the heaviest at the bottom and nearest the front, which will not only lower the point 0, but will bring it for- ward, and nearer the proper position m. Part of the load, too, might be suspended below the fore part of the carriage in dry weather, and the centre of gravity would approach still nearer the point m. When the point m is thus depressed, the weight on the horse is not only judiciously regulated, but the cart will be prevented from overturning, and in rugged roads the weight sus- tained by each wheel will be in a great degree equalized. In loading four-wheeled carriages, great care should be taken not to throw much of the load upon the fore wheels, as they would otherwise be forced deep into the ground, and require great force to pull them forward. In some modern carriages this is very little attended to. The coachman's seat is sometimes enlarged so as to hold two persons, and all the baggage is gene- rally placed in the front directly above the fore- wheels. By this means, the greatest part of the load is upon the small wheels, and the draught becomes doubly severe for the poor animals, who must thus unnecessarily suffer for the ignorance and folly of man. MECHANICS. ON THE THRASHING MACHINE. IN a country like this, where agriculture has utility of arrived at such a high state of perfection, thejJJ^J utility of thrashing machines cannot easily be ing labour called in quesion. The universal prevalence of these engines is a strong proof that they are advantageous to the farmer; and, however much some men may inveigh against the adop- tion of every kind of machinery that has for its object the abridgement of manual labour, yet we are convinced, that no evil consequences can possibly accrue from their introduction; and that such insinuations have a tendency tQ inflame the minds of the vulgar, and retard the progress of science. As a proof of this, we might mention the fate of the celebrated Ark- wright, the inventor of the fly-shuttle, whom the fury of an English rabble banished from his na- tive country. The thrashing machine was invented in Scot- History of land, in 1758, after five years labour, by Mv^*' Michael Stirling, a farmer in Perthshire. Thecbine. honour of this invention has been claimed by Mr. Andrew Meikle, an ingenious mill-wright in East Lothian, who obtained a patent for one Vol. IL X 322 MECHANICS. of these machines, about the year 1785 ; and in this country his claims have been generally ad- mitted. Mr. Meikle, however, was merely an improver of the thrashing machine, and I am assured by a gentleman of the most unquestion- able authority, who, from his local situation, had access to the best information, that Mr. Meikle had seen Mr. Stirling's thrashing ma- chine before he erected any of his own, and that he merely altered and improved it. About 26 years prior to the date of Mr. Stirling's inven- tion, a thrashing machine was constructed in Edinburgh, by Mr. Michael Menzies, which ope- rated by the elevation and depression of a number of flails, by means of the motion of a crank ; and, in 1 767, the model of a thrashing mill, invented by Mr. Evers of Yorkshire, was laid before the Society of arts in London, who rewarded the inventor with a premium of 6O pounds. This machine, which was driven by wind, consisted of a number of stampers, that beat out the grain when laid upon a moveable thrashing floor, and was actually used on a large scale in Yorkshire, where it received the approbation of several intelligent gentlemen of the county. 1 All these machines, however, and others of a similar kind, with which the public are perpetually harassed, are completely defec- tive in principle, and are greatly inferior to the worst of those now in use, which operate by the revolution of a thrashing scutch furnished with beaters, the exclusive invention of our country- man Mr. Stirling. 1 Bailey's Drawings of Machines laid before the Society of Art8,^vol. i, pp. 54-59. MECHANICS* 323 On Thrashing Machines driven by Water. In Fig. 1 , Plate VII, AE is an undershot water Plate vn, wheel, which drives the machinery. On its axis Flg- x- is fixed the spur- wheel CD, furnished with 150 c r g teeth, which impel the pinion b, containing 25 driven by. teeth. On the axis H of the pinion b is plac- watcr> ed another wheel E, carrying 72 teeth, which take into the 15 leaves of the pinion c. The axle xx of the thrashing scutch, represented more distinctly in Fig. 2, by yxy, is fastened upon the same axis with the pinion c, and is therefore carried round with the same velocity. The thrashing scutch, a section of which may be seen in Fig. 3, is generally furnished with Fig. ;. four, and sometimes with a greater number of beaters, yij, whose surfaces, o, o, are covered with iron rounded off at the edges, in order to prevent them from cutting the straw. When, these beaters strike upwards, the scutch must be contained in a hollow cylinder of wood mn, so that the tops y, y, o, o, of the beaters may- be above it ; in which case, the scutch is called the thrashing drum. But when the beaters strike downwards, there is no occasion for covering it with boards. The gudgeon of the axis H carries a wheel i of 22 teeth, which acts upon the wheel h with 1 8 teeth ; on the axis h e is fixed another wheel e with 1 7 teeth, that drives the crown wheel d, furnished with three rows of teeth, 13, 17, and 21, which, by means of the spindle R, gives motion to one of the feeding rollers, not visible in Fig. 1, but represented distinctly by RR in Fig. 2. On the axis of the upper feeding roller RR is placed a small pinion, which drives the X2 324 MECHANICS. under feeding roller by acting upon another pinion, with the same number of teeth, fixed upon its spindle. The two feeding rollers, which are generally 3-f inches in diameter, are fluted, or cut into small leaves like pinions, so that the leaves of the one may take into the leaves of the other ; and their gudgeons move in mortises of such a nature, that the upper roller may rise in its frame, and the under one remove from the beaters, when too much corn is admitted between them. In order that the velocity of the rollers may be increased and diminished at pleasure, according to the nature of the corn to be thrashed, the wheel e is made to shift on its axis so as to act upon any of the three rows of teeth in the crown wheel d, which enable us to communicate three different degrees of velocity to the rollers. As the machinery which drives the straw- shaker interferes, in Fig. 1, with that which gives motion to the fluted rollers, it will be seen fig. a. in Fig. 2, which is a plan of the machine where the corresponding parts are marked by similar letters. The wheels b, E, c 9 in Fig. 1, are not represented in this figure, but H is the extremi- ty of the axle on which E and b are fixed. The small wheel i of 22 teeth, fixed upon the extre- mity of the gudgeon i H gives motion to m, a wheel of 17 teeth, which, by the intervention of the spindle m n and wheel n of 24 teeth, drives o, a wheel carrying 34 teeth. * On the same 1 The dimensions of the thrashing machines here de- scribed are chiefly taken from Gray's Experienced Mill- wright, a book of great utility in a manufacturing country. It consists chiefly of plans, sections, and elevations, of dif- ferent machines, which the author himself has either erected, or whose construction he has immediately super- intended. We are afraid, however, that Mr. Gray has 3 MECHANICS. 325 axis with o is fixed the straw-shaker KK S on whose cross arms are fastened the rakes z r, fur- nished with a number of iron, or wooden, teeth, which carry off the straw, while tfre grain falls down into the fanners. The axis of these fan- ners p q, Fig. 1, is put in motion by the belt pp passing over the two rollers />, p. A section of the straw-shaker is shewn in Fig. 4, where K is its axle, z, z, its arms, and r, r, &c. the teeth fastened at the extremity of these arms. That the reader may have a distinct idea ofy eloc!t y f the thrashing machine, we have calculated the ^rts. c ' following table, which exhibits the number of teeth in the wheels, and the velocity of its dif- ferent parts. It is scarcely necessary to premise, that when one wheel drives another, the num- ber of turns, or parts of a turn, performed by the wheel which is driven, is represented by a fraction, whose numerator is the number of teeth in the wheel that gives the motion, and whose denominator is the number of teeth in the wheel which receives it. Thus, a wheel with 25 teeth, driven by another with 15O, will perform Vy , or six revolutions for one revolution of the im- pelling wheel; and a wheel with 16 teeth, driven by a pinion with 8 teeth, will make ~, or y of a turn for one revolution of the pinion. When two or more wheels are upon the same axis, they all perform the same number of revo- lutions, however different be their magnitude not rendered these machines sufficiently intelligible to the uninstructed mechanic, from the great brevity of his de- scriptions ; and, we hope, if his work reaches a second edition, as we trust it will, that he will take advantage of this friendly hint. Xs 326 MECHANICS. and the number of teeth ; though the velocities of their circumferences may be widely different. In the following table, we have calculated merely the number of turns made by each wheel for one turn of the water wheel ; but when the number of revolutions performed by the water wheel in a second is known, we have only to multiply the quantities in the third column by this number, in order to find the number of turns which each wheel makes in the same time. Names of the wheels. Number of Number of turns teeth in each for one of the water- wheel, i wheel. Plate VII, Fig. i & a. Teeth. Turns. Dec. ^jQ 150 1.000 ft 25 6.00Q - E 72 6.00O c 15 28.800 Thrashing-scutch, 28.OOO i 22 6.OOO h 18 7.333 e 17 7.333 Cd 13 9.534 Fluted Rollers 1 d 17 7.333 (.d 21 5.94O m 17 7.764 n 24 7.764 34 5.479 traw-shaker. 5.479 The working parts of the thrashing machine being thus described, the manner of its operation will be easily understood. The sheaves of corn are spread upon an inclined board 0, called the feeding board, and introduced between the flut- MECHANICS. 327 ed rollers, a section of which is distincly visible at ii, in Figure 3. The corn is held fast p ! ateV ' by these rollers, which are only about three lg ' 3 ' quarters of an inch from the beaters, while the thrashing drum, or scutch, revolving with im- mense rapidity and force, separates the grain from the straw, by the repeated strokes of the beaters. Part of the grain falls through the heck or scarce i r, into a large hopper, which conducts it to the fanners, and some of it is car- ried along with the straw into the other heck r p, where it falls into the hopper, while the straw is cleared away by the rakes z of the straw-shaker, and thrown out at the opening np into the lower part of the bam. In some thrashing machines driven by water, the motion is conveyed to the thrashing scutch* by means of a long perpendicular axis. The lower extremity of the axis is furnished with a pinion, which is driven by a spur-wheel, with teeth perpendicular to its plane, placed upon the axis of the water wheel. A large horizontal wheel is fixed on the top of this long axle, which acts upon a pinion fastened upon the axis of the thrashing-drum. On Thrashing Machines driven by Horses. Wherever a sufficient quantity of water can Thrashing be procured, it should always be employed asJJJ^J* the impelling power of thrashing machines, horses. There are many situations, however, in which it cannot be obtained ; and as the erection of steam-engines and wind-mills would be too ex- pensive for the generality of farmers, they are under the necessity of having recourse to animal Plate vnr. power. In Plate VIII, Fig. 1, is represented a lg ' x 328 MECHANICS. thrashing machine, which may be driven by four or six horses. To the vertical axis M six strong bars are fixed, called the horse polls, four of which, P, /2, 9, Z/, are visible in the figure, and to the extremity of each of these poles two pieces of wood, like op, are attached, to which the horses are yoked when the machine is to be used. Upon the top of the six poles is placed the large bevelled wheel AB^ containing 27O teeth, which drives the pinion BC of 40 teeth ; on the axle N is also fixed the wheel DD, which carries 84 teeth, and drives the pinion b of 24 teeth, placed upon the axle b k. Upon the same axis the wheel EE revolves, carrying 66 teeth, which drive the pinion c of 1 5 teeth, and conse- quently the thrashing-drum xx, which is fixed npon the same axle. The feeding rollers are driven by the intervention of the four bevelled wheels 2, k, e, d, the latter of which is fast- ened on the axis of the upper feeding roller. The wheel f, upon the gudgeon i b, contains 25 teeth, the wheel h 24 teeth, e 22 teeth, and d 21 teeth ; but when the fluted rollers require a greater velocity, e is taken from its iron axle, and a greater or less wheel substituted in its room. The short axle b k is furnished with a pulley />, which, by means of the leathern belt pp, gives motion to the fanners placed below the thrashing scutch and straw- shaker. Fig. a. Fig. 2 represents a plan of the wheels, thrash- ing-drum, and straw-shaker, where the corre- sponding parts, in Fig. 1, are marked with simi- lar letters. The small wheels g and k, however, which convey motion to the straw-shaker, are not seen in Fig. 1 . The largest one g is fixed on the axis N, and carries 38 teeth. It drives k> which contains 1 4 teeth, and is placed upon the axis of the straw-shaker KK. MECHANICS. 329 An elevation of the working parts of the ma- chine is delineated in Fig. 3, where the corre- Fig. spending parts in the plan and section have the same letters affixed to them. The sheaves of corn are spread on the feeding board o, drawn in by the rollers i, z, and thrashed by the beaters 0, 0, which strike downward. Part of the corn falls through the heck i r, and some part of it is carried along with the straw into the larger heck rp, where it falls into the hopper below, while the straw is thrown out at the opening np. The drum and straw-shaker are surrounded with a covering of wood i m n. The following table exhibits, at one view, the number of teeth in the wheels, and the different velocities with which they move. Names of the wheels. Number of teeth in each wheel Number of turns for one of the wheel. Plate VIII, Fig. i, 3, 4. Teeth. Turns. Dec. AE 270 1.000 EC 40 6.750 DD 84 6.750 b 24 23.625 EE 66 23.625 c 15 1O3.950 Thrashing-scutch. 8 k 38 14 103.95O 6.750 18.293 Straw-shaker. 18.293 i 25 23.625 h 24 24.617 e 22 26.857 d Feeding Rollers. 21 28.199 28.199 33O MECHANICS. Driven by In situations where there is an occasional sup- vrateF. ply of water, thrashing machines are sometimes constructed so as to be driven either by horses or -water. In this case, the water-wheel has the position LH 9 Fig. 1, and is furnished with a large wheel GH consisting of segments of cast iron firmly fixed to the arms of the water-wheel. The wheel GH drives FG, and thus communi- cates motion to the horizontal shaft JV, and the rest of the machinery. When there is no water for impelling the mill, the water-wheel LH is either lowered in its frame, or one of the seg- ments is taken from the wheel GH, in order to keep it clear of the wheel FG; and when there is a sufficient discharge of water CB is either raised above AB, or AB is deprived of a few teeth, which can be screwed and unscrewed at pleasure. Sometimes, when there is a small sup- ply of water, its energy may be combined with the exertion of one or two horses. Driven by if the thrashing machine is to be driven by sleam. r wind, the motion is conveyed to the axle N 9 by the small wheel mC 9 fixed at the bottom of the vertical axis w, which is moved by the wheel upon the windshaft. If the mill is to be moved by steam, the large fly must be fixed on the axis N, parallel to the horizon. Fig. 4. Fig. 4 represents a thrashing machine of a very simple construction, which may be driven by two or three horses. The large wheel and -*: pinion, corresponding with AB and C, in Fig. 1 , are not delineated in the figure, but the former contains 166, and the latter 19, teeth. On the shaft N is fastened the wheel DD ? which carries 80 teeth, and drives the pinion c of 9 teeth, and consequently the thrashing-drum, which is fixed on its axis. The strauvshaker is Burned bv means of the leathern belt h i pass- MECHANICS. 331 ing over the pulleys h and /, the fluted rollers by the belt k m, and the fanners by the rope d e. I have seen a thrashing mill of this simple construction belonging to Hercules Ross, Esq. of Rossie, which was driven by six horses. Thepi.AT large wheel, corresponding with AB> had 144^. m> j teeth, and the pinion, corresponding with C, had 14 teeth. The wheel DD had 8O teeth, and the pinion c 8. The straw-shaker and fluted rollers were driven by belts, and the fanners by a rope passing over a groove in the large wheel DD. The thrashing-drum revolved 103 times for every turn of the horses ; whereas the drum in the machine, represented in Fig. 4, performed only 7Q revolutions in the same time. In the first case, however, the horse walk was of such a size that the horses performed only 3 turns in a minute ; while, in the latter, the horses are supposed to make 4 revolutions in a minute. The velocity, therefore, of the former will be 4 x 79 n: 3 16, and the velocity of the latter 3X103=309. When thrashing mills began to be generally adopted in this country, they were constructed according to the plan represented in Fig. S.Fig.j. The wheel AE has 276 cogs, b 14, the crown wheel c 84, d 16. The thrashing-drum is fixed on the axis m d, and the fanners, straw- shaker, and fluted rollers, are moved by leathern belts. A thrashing machine for small farms, which A sma .H can be wrought by a single horse, has long been a desideratum in mechanics, and every attempt to construct one on a small scale seems to have completely failed. While examining tfje causes of this failure, I have thought of Spme methods by which they may be partially 332 MECHANICS. removed, and of a machine which might be im- pelled by one horse, or by two or three men working at a winch. The description of this simple engine I expected to have communi- cated in this article ; but a desire to im- prove it as much as possible, has induced me to defer its publication to some future oppor- tunity. On the Power of Thrashing Machines. Power of The quantity of corn which a machine w ^ thrash m a given time, depends so much upon the judicious formation and position of its parts, that one machine will often perform double the work of another, though construct- ed upon the same principles, and driven by the same impelling power. Misled by this cir- cumstance, those who have given an account of the power of their thrashing mills, have published merely the number of bolls which they can thrash in a given time, without mentioning the quantity of impelling power, or the number of horses employed to drive them. Mr. Fenwick, whose labours in practical me- chanics we have already mentioned with com- mendation, has furnished us with some import- ant information upon this point. He found, from a variety of experiments, that a power capable of raising a weight of JOOO pounds, with a velocity of 15 feet per minute, will thrash two bolls of wheat in an hour ; and that a power sufficient to raise the same weight, with a velocity of 22 feet per minute, will thrash three bolls of the same grain in an hour. From these facts, Mr. Fenwick has com- MECHANICS. 333 puted the following table, which is applicable to machines that are driven either by water or horses. ''fable of the Power of Thrashing Machines. Gallons of water per minute, ale measure, Gallons of water per minute, ale measure, Gallons of water per minute, ale measure, Num- ber of horses work- Bolls of wheat thrash- Bolls thrashed in 9$ hours discharged on an overshot wheel 10 feet in discharged on an overshot wheel ij feet in discharged on an overshot wheel ao feet in hours. ed in an hour. actual working, or in a day. diameter. diameter. diameter. 230 160 130 1 2 19 390 296 205 2 3 28J- 528 380 272 3 5 47-r 660 470 340 4 7 66f 790 565 400 5 9 85-j- 970 680 50O 6 10 95 1 2 3 4 5 6 The four first columns of the preceding table contain different quantities of impelling power, and the two last exhibit the number of bolls of wheat in Winchester measure, which such powers are capable of thrashing in an hour, or in a day. Six horses, for example, are capable of thrashing 10 bolls of wheat in an hour, or 95 in the space of 97 hours, or a working day ; and 680 gallons of water discharged during a minute into the buckets of an overshot water wheel 15 feet in diameter, will thrash the same quantity of grain. MECHANICS. ON THE NATURE OF FRICTION, AND THE METHOD OF DIMINISHING ITS EFFECTS IN MACHINERY. importance J. HE resistance which friction generates in the in C the d na cornmun icating parts of machinery is so power- ture and ful, and the consequent defalcation from the im- friction 0f P e ^ m g power is so great, that a knowledge of its nature and effects must be of the highest im- portance to the philosopher and the practical mechanic. The theory of mechanics must con- tinue imperfect till the nature and effects of fric- tion are thoroughly developed, and their per- formance must be comparatively small, and the expence of their erection and preservation com- paratively great, till some effectual method is dis- covered for diminishing that retardation of the machine's velocity, and that decay of its materi- als which arise from the attrition of the connect- ing parts. The knowledge, however, which has been acquired concerning this abstruse subject has not been commensurate with the labours of MECHANICS. 335 philosopher and the progress of other branches of mechanical science ; and those contrivances which ingenious men have discovered for dimi- nishing the resistance of friction, have either been overlooked by practical inquirers, or rejected by those vulgar prejudices which prompt the me- chanics of the present day to persist in the most palpable errors, and neglect such maxkns of construction as are authorized both by theory and experience. It may be proper, therefore, in a work like this, to give a summary view of the opinions of different philosophers upon the nature of friction, and the means which may be adopted for diminishing its effects. M. Amontons was the first philosopher who favoured us with any thing like correct informa- tion upon this subject. He found that the resist- ance opposed to the motion of a body upon a horizontal surface was exactly proportional to its weight, and was equal to one third of it, or more generally to one third of the force with which it \vas pressed against the surface over which it moved. He discovered also that this resistance did not increase with an increase of the rubbing superficies, nor with the velocity of its motion. * The experiments of M. Bulfinger authorized conclusions similar to those of Amontons, with this difference only, that the resistance of fric- tion was equal only to one fourth of the force with which the rubbing surfaces were pressed together. * 1 Mem. de 1'Acad. Paris, 1699, p. 2O6. Amonton's experiments were confirmed by Bossut and Belidor. See Architect. Hydraulique, voL 1, chap, ii, p, 70- 2 Comment. Petropol. torn, ii, p. 40. 336 MECHANICS. This subject was also considered by Parent, who supposed that friction is occasioned by small spherical eminences in one surface being dragged out of corresponding spherical cavities in the other, and proposed to determine its quantity by finding the force which would move a sphere standing upon three equal spheres. This force was found to be to the weight of the sphere as 7 to 20, or nearly one third of the sphere's weight. 3 In investigating the phenomena of fric- tion, M. Parent placed the body upon an inclin- ed plane, and augmented or diminished the angle of inclination till the body had a tendency to move ; and the angle at which the motion com- menced, he called the angle of equilibrium. The weight of the body, therefore, will be to its fric- tion upon the inclined plane, as radius to the sine of the angle of equilibrium, and its weight will be to the friction on a horizontal plane, as radius to the tangent of the angle of equilibrium. 4 The celebrated Euler seems to have adopted the hypothesis of Bulfinger respecting the ratio of friction to the force of pression ; and in two curious dissertations which he has published up- on this subject, 5 has suggested many important observations, which have been of great use to future enquirers. He observes, that when a body is in motion, the effect of friction will be only one half of what it is when the body has 5 Recherches de Mathematique et Physique, 1713, torn, ii, p. 462. 4 Mem. de I'Acad. Paris, 1704, p. 174. * The first is entitled, Sur le frottement des Corps solicits, and the other, Sur la diminution de la resistance du frottement, published in the Mem. de I'Acad. Berlin, ann. 1748, pp. 122, 133. " " - MECHANICS. S3? begun to move ; and he shews that if the angle of an inclined plane be gradually increased, till the body which is placed upon it begins to de- scend, the friction of the body at the very com- mencement of its motion will be to its weight or pressure upon the plane, as the sine of the plane's elevation is to its cosine, or as the tan- gent of the same angle is to radius, or as the height of the plane is to its length. But when the body is in motion the friction is diminish- ed, and may be found by the following equa- tion F ir Tan. a in which .Fis 15625 nn cos. fl, the quantity of friction, the weight or pressure of the body being 1 ; a the angle of the plane's inclination, m the length of the plane in 100O th parts of a rhinland foot, and n the time of the body's descent. Respecting the cause of friction, Euler is nearly of the same opinion with Parent : the only difference is, that instead of re- garding the eminences and corresponding depres- sions as spherical, he supposes them to be angu- lar, and imagines the friction to arise from the body's ascending a perpetual succession of inclin- ed planes. Mr. Ferguson found that the quantity of fric- tion was always proportional to the weight of the rubbing body, and not to the quantity of sur- face, and that it increased with an increase of ve- locity, but was not proportional to the augmen- tation of celerity. He found also that the fric- tion of smooth soft wood, moving upon smooth soft wood, was equal to y of the weight ; of rough wood upon rough wood -7 of the weight ; of soft wood upon hard, or hard upon soft, 7 of the weight ; of polished steel upon polished steel or pewter -J- f the weight j of polished steel up- Fol. II. Y 335 MECHANICS. on copper f, and of polished steel upon brass -g-, of the weight. 6 The Abbe Nollet 7 and Bossut 8 have distin- guished friction into two kinds ; that which arises from one surface being dragged over another, and that which is occasioned by one body rolling upon another. The resistance which is generat- ed by the first of these kinds of friction is always greater than that which is produced by the se- cond ; and it appears evidently from the experi- ments of Muschenbroek, Schoeber, and Meis- ter, that when a body is carried along with an uniformly accelerated motion, and retarded by the first kind of friction, the spaces are still pro- portional to the squares of the times, but when the motion is affected by the second kind of fric- tion, this proportionality between the spaces and the times of their description does not obtain. Result of The subject of friction has more lately occu- vmce's ex- p j e( j t j le attention of the ingenious Mr. Vince of penments. *L ,., TT r 111 c rt_ i Cambridge. He round that the rnction or hard bodies in motion is an uniformly retarding force, and that the quantity of friction, considered as equivalent to a weight drawing the body back- ^, wards, is equal to M - -- where M is r t the moving force expressed by its weight, W the weight of the body upon the horizontal plane, S the space through which the moving force or weight descends in the time t and r ==. 16.087 feet, the force of gravity. Mr. Vince also found * Tables and Tracts, edit. 2d, p. 289. 7 Nollet, Lecons de Physique, torn, iii, p. 231, ed. 1770. ' Traite Elementaire de Mecanique, par Bossut, 306-7. MECHANICS. 339 that the quantity of friction increases in a less ra- tio than the quantity of matter or weight of the body, and that the friction of a body does not continue the same when it has different surfaces applied to the plane on which it moves, but that the smallest surfaces will have the least fric- tion. 9 Notwithstanding these various attempts to un- fold the nature and effects of friction, it was re- served for the celebrated Coulomb to surmount the difficulties which are inseparable from such an investigation, and to give an accurate and sa- tisfactory view of this complicated part of mecha- nical philosophy. By employing large bodies and ponderous weights, and conducting his experi- ments on a large scale, he has corrected several errors which necessarily arose from the limited experiments of preceding writers; he has brought to light many new and striking phenomena, and confirmed others which were hitherto but partial- ly established. As it would be foreign to the na- ture of this work to follow Monsieur Coulomb through his numerous and varied experiments, we shall only present the reader with the new and interesting results which they authorize. 1 1. The friction of homogeneous bodies, or bo- Friction of dies of the same kind moving upon one another, hom g ?. ne " o r ous bodies is generally supposed to be greater than that or not always 9 Philosophical Transactions, v. Ixxv, p. 167. 1 A full account of Coulomb's experiments may be seen in the Journal de Physique for September and October 1785, vol. xxvii, pp. 206 & 282, &c. An excellent summary of them may also be found in Van Swinden's Positiones Physicae. They were originally published in the Memoires des Savons f rangers, torn, x, p. 163 ; and obtained the (double prize offered by the Academy in 1779 and 1781. Y 2 MECHANICS. greater heterogeneous bodies ;* but Coulomb has shewn than that that there are exceptions to this rule. He found, gene^S " for example, that the friction of oak upon oak bodies. wa s equal to j^-j of the force of pression ; the friction of pine against pine was 7^7, and of oak against pine . The friction of oak against copper was -^, and that of oak against iron near- ly the same. 3 Friction 2. It was generally supposed, that in the case fording to" f wo dj the friction is greatest when the bodies the course are dragged contrary to the course of their fibres 6 fibres ; 4 but Coulomb has shewn that the fric- tion is, in this case, sometimes the smallest. When the bodies moved in the direction of their fibres the friction was -^ of the force with which they were pressed together; but when the motion was contrary to the course of the fibres, the friction was only -5. Friction in- 3. The longer the rubbing surfaces remain wSTthe in contact J the greater is their friction. 5 When time of wood was moved upon wood, according to the contact, direction of the fibres, the friction was increased by keeping the surfaces in contact for a few se- conds ; and when the time was prolonged to a minute, the friction seemed to have reached its utmost limit. But when the motion was per- * This was the opinion of Muschenbroek, Krafft, Ca- mus, and Bossut. J From a series of experiments on heavy machinery where the force of pression was about 33 cwt. Mr. Southern of Birmingham concludes, that in favourable cases, the friction does not exceed -^ of the force of pression. It is to be wished that this curious result were confirmed by other writers. 4 Muschenbroek, Introductio ad Philosoph. Nat. $513. J This is mentioned by Bossut, Traite de Mecanique, 310 ; but Coulomb has the merit of having established the fact. MECHANICS* 341 formed contrary to the course of the fibres, a greater time was necessary before the friction arrived at its maximum. When wood was moved upon metal, the friction did not attain its maxi- mum till the surfaces continued in contact for 5 or 6 days; and it is very remarkable, that when wooden surfaces were anointed with tal- low, the time requisite for producing the great- est quantity of friction was increased. The in- crease of friction which is generated by prolong- ing the time of contact is so great, that a body weighing 1650 pounds was moved with a force of 64 pounds when first laid upon its corres- ponding surface. After having remained in contact for the space of 3 seconds, it required 16O pounds to put it in motion, and when the time was prolonged to 6 days, it could scarcely be moved with a force of 622 pounds. When the surfaces of metallic bodies were moved up- on one another, the time of producing a maxi- mum of friction was not changed by the inter- position of olive oil ; it was increased, however, by employing swines grease as an unguent, and was prolonged to 5 or 6 days by besmearing the surfaces with tallow. 4. Friction is in general proportional to the Friction force with which the rubbing surfaces are press- a]^ ed together ; and is, for the most part, equal to force of between f and 7 of that force. 6 In order to f ns * D - prove the first part of this proposition, Coulomb employed a large piece of wood, whose surface * Friction being -J. of the force of pression, it may be shewn, that a power which moves a body along a horizon- tal plane, acts to the greatest advantage when the line of direction makes an angle of 1 8 2& with the plane. Thi proposition is neatly demonstrated in Mr. Gregory's tyfe- chanics, vol. ii, p. 1 ?. YS 342 MECHANICS. contained 3 square feet, and loaded it succes- sively with 74 pounds, 874 pounds, and 2474 pounds. In these cases the friction was succes- sively -qs, ~, 5^, of the force of pression; and when a less surface and other weights were used, the friction was ^ 6 , ~ 9 ~. Similar re- sults were obtained in all Coulomb's experi- ments, even when metallic surfaces were em- ployed. The second part of the proposition has also been established by Coulomb. He found that the greatest friction is engendered when oak moves upon pine, and that it amounts to ~^ of the force of pression ; on the contrary, when iron moves upon brass, the least friction is produced, and it amounts to ~ of the force of pression. Frictionnot 5. Friction is in general not increased by aug- aftoThe" 1 menting the rubbing surfaces. 7 When a super- rubbing ficies of 3 feet square was employed, the friction, with different weights, was ~ at a medium ; but when a smaller surface was used, the friction, instead of being greater, as might have been ex- pected, was only ^. Friction g. Friction for the most part is not augment- sometimes j , r i v T diminished e & by an increase or velocity. In some cases, byincreas. however, it is diminished by an augmentation j;V heve "of celerity. 8 M. Coulomb found, that when 7 Muschenbroek and Nollet entertained the opposite opinion. The experiments of Krafft coincide with those of Coulomb. See Commen. Fetropol. torn, xii, p. 266, 19, 20, &c. 8 The latter part of this proposition is confirmed by a circumstance which occurred in the course of M. Lam- bert's experiments on undershot mills, but which he im- putes to a very different cause. He found that the resist- ance which is generated by the friction of the communi- cating parts of a corn mill, combined with that which arises from the grain between the mill-stones, always diminished when the velocity was increased. M. Lambert did not hesitate MECHANICS. wood moved upon wood in the direction of the fibres, the friction was a constant quantity, how- ever much the velocity was varied ; but that when the surfaces were very small, in respect to the force with which they were pressed, the friction was diminished by augmenting tlie rapidi- ty : the friction, on the contrary, was increased when the surfaces were very large when com- pared with the force of pression. When the wood was moved contrary to the direction of its fibres, the friction in every case remained the same. If wood is moved upon metals, the fric- tion is greatly increased by an increase of velo- city; and when metals move upon wood be- smeared with tallow, the friction is still aug- mented by adding to the velocity. When metals move upon metals, the friction is always a con- stant quantity ; but when heterogeneous sub- stances are employed which are not bedaubed with tallow, the friction is so increased with the velocity, as to form an arithmetical progression when the velocities* form a geometrical one. 7. The friction of loaded cylinders rolling Friction of upon a horizontal plane, is in the direct ratio of J? a< ? ed c ?~ tf . . , 11- r i T hnders. their weights, and the inverse ratio or their dia- meters. In Coulomb's experiments, the friction of cylinders of guaiacum wood, which were two inches in diameter, and were loaded with 100O pounds, was 18 pounds or ~ of the force of hesitate to assert, that the part of this compound resistance which was produced by the friction of the machinery con- tinued invariably the same, and ascribed, without any reason, the diminution which accompanied an increase of velocity to a diminution of the grain's resistance between the mill-stones ; whereas it was probably a diminution of the friction of the connecting parts, occasioned ,by the augmentation #f their velocity. 344 MECHANICS. pression. In cylinders of elm, the friction was greater by T , and was scarcely diminished by the interposition of tallow. Friction of From a variety of experiments on the friction pulleys. 80 f tne axes f pullies, Coulomb obtained the following results. When an iron axle moved in a brass bush or bed, the friction was ^ of the pression ; but when the bush was besmeared with very clean tallow, the friction was only -^ ; when swines grease was interposed, the fric- tion amounted to ^ ; and when olive oil was employed as an unguent, the friction was never less than or ^. When the axis was of green oak, and the bush of guaiacum wood, the fric- tion was ~ when tallow was interposed ; but when the tallow was removed so that a small quantity of grease only covered the surface, the friction was increased to ~. When the bush was made of elm, th friction was in similar circumstances T T T and ^-, which is the least of all. If the axis be made of box, and the bush of guaiacum wood, the friction will be ~ and ~, circumstances being the same as before. If the axle be of boxwood, and the bush of elm, the friction will be and ^ ; and if the axle be of iron, and the bush of elm, the friction will be ^ of the force of pression. Having thus given a brief, though we trust a comprehensive view of the interesting results of Coulomb's experiments, we shall conclude this part of the subject, by presenting the reader with some excellent and original observations on the cause of friction, by Mr. John Leslie, Professor of Mathematics in the university of Edinburgh. 7 7 See his ingenious and profound work on the Nature and propagation of heat, chap, xv, p. 299, &c. MECHANICS. 345 c If the two surfaces which rub against each ^^ ** * other are rough and uneven, there is a neces- n ' sary waste of force, occasioned by the grind- * ing and abrasion of their prominences. But * friction subsists after the contiguous surfaces * are worked down as regular and smooth as ' possible. In fact, the most elaborate polish ' can operate no other change than to diminish ' the size of the natural asperities. The surface ' of a body, being moulded by its internal struc- ' ture, must evidently be furrowed, or toothed, ( or serrated. Friction is, therefore, common- ' ly explained on the principle of the inclined 6 plane, from the effort required to make the 4 incumbent weight mount over a succession of * eminences. But this explication, however cur- 4 rently repeated, is quite insufficient. The mass * which is drawn along is not continually ascend- 6 ing ; it must alternately rise and fall ; for each ' superficial prominence will have a correspond- 4 ing cavity ; and since the boundary of con- ' tact is supposed to be horizontal, the total ele- ' vations will be equalled by their collateral de- ' pressions. Consequently, if the lateral force e might suffer a perpetual diminution in lifting ' up the weight, it would, the next moment, ' receive an equal increase by letting it down ' again; and those opposite effects, destroying * each other, could have no influence whatever ' on the general motion. e Adhesion seems still less capable of account- ' ing for the origin of friction. A perpendicu- ' lar force acting on a solid can evidently have ' no effect to impede its progress ; and though ' this lateral force, owing to the unavoidable c inequalities of contact, may be subject to a 6 certain irregular obliquity, the balance of 346 MECHANIC*. * chances must, on the whole, have the same ' tendency to accelerate, as to retard, the mo- c tion. If the conterminous surfaces were, there- e fore, to remain absolutely passive, no friction * could ever arise. Its existence demonstrates c an unceasing mutual change of figure, the c opposite planes, during the passage, continual- c ly seeking to accommodate themselves to all ' the minute and accidental varieties of contact. ' The one surface, being pressed against the 4 other, becomes, as it were, compactly indent- ' ed, by protruding some points and retracting ' others. This adaptation is not accomplished 6 instantaneously, but requires very different e periods to attain its maximum, according to the e nature and relation of the substances concern- 6 ed. In some cases, a few seconds are suf- * ficient, in others, the full effect is not pro- ' duced till after the lapse of several days. ' While the incumbent mass is drawn along, at * .every stage of its advance, it changes its ex- * ternal configuration, and approaches more or e less towards a strict contiguity with the under ' surface. Hence the effort required to put it * first in motion, and hence too, the decreased ' measure of friction, which, if not deranged ' by adventitious causes, attends generally an ' augmented rapidity. This appears clearly ' established by the curious experiments of e Coulomb, the most original and valuable which f have been made on that interesting subject. 4 Friction consists in the force expended to 6 raise continually the surface of pressure by ' an oblique action. The upper surface travels ' over a perpetual system of inclined planes ; f but that system Is ever changing, with alter- * nate inversion. In this act, the incumbent MECHANICS. 347 w weight makes incessant, yet unavailing efforts ' to ascend : for the moment it has gained the 6 summits of the superficial prominences, these * sink down beneath it, and the adjoining cavi- ' ties start up into elevations, presenting a new 6 series of obstacles which are again to be sur- * mounted ; and thus the labours of Sisiphus are * realized in the phenomena of friction. * The degree of friction must evidently de- * pend on the angles of the natural protuber- * ances, which are determined by the element- * ary structure or the mutual relation of the * two approximate substances. The effect of ' polishing is only to abridge those asperities ' and increase their number, without altering ' in any respect their curvature or inflections. ' The constant or successive acclivity produced ' by the ever varying adaptation of the conti- 4 guous surfaces, remains, therefore, the same, * and consequently the expence of force will ' still amount to the same proportion of the c pressure. The intervention of a coat of oil, e soap, or tallow, by readily accommodating * itself to the variations of contact, must tend ' to equalize it, and therefore must lessen the ' angles, or soften the contour, of the succes- ' sively emerging prominences, and thus dimi- ' nish likewise the friction which thence re- ' suits.' Having thus considered the origin, the na- Method of ture, and the effects, of friction, we shall now f n im ^ e sh e " f . attend to the method of lessening the resistance fccts of which it opposes to machinery. The most ef- friction - ficacious mode of accomplishing this, is to con- vert that species of friction which arises from one body being dragged over another, into that which is occasioned by one body rolling upon 346 MECHANICS. another. As this will always diminish the resist- ance, it may be easily effected by applying wheels or rollers to the sockets or bushes which sustain the gudgeons of large wheels, and the axles of wheel carriages. Casatus 8 seems to have been the first who recommended this apparatus. Friction It was afterwards mentioned by Sturmius 9 and wheels. Wolfius ;" but was not used in practice till Sully* applied it to clocks in the year 1716, and Mondran 3 to cranes in 1725. Notwith- standing these solitary attempts to introduce friction wheels, they seem to have attracted but little attention till the celebrated Euler ex- amined and explained, with his usual accuracy, their nature and advantages. 4 The diameter of the gudgeons and pivots should be made as small as the weight of the wheel and the impel- ling force will permit. The gudgeons should rest upon two wheels as large as circumstances will allow, having their axes as near each other as possible, but no thicker than what is abso- lutely necessary to sustain the superincumbent weight. When these precautions are properly attended to, the resistance which arises from the friction of the gudgeons, &c. will be extremely trifling. 5 1 Mechan. lib. ii, cap. i. p. 130. 9 Misccllan. Berolinens. torn, i, p. 305. 1 Opera Mathematica, torn, if, p. 684. 1 Machines approuvees, torn. No. 177- 3 Id. No. 254. 4 Mem. de 1'Acad. Berlin 1748, p. 133. J Mr. Walker, a lecturer on Experimental Philosophy, has boldly pronounced friction wheels to be ' expensive * nonsense,' (System of Famil. Philos. v. i.) This gen- tleman should have recollected that they were recommend- ed by Euler and many distinguished philosophers ; and, though MECHANICS. The effects of friction may likewise in some Friction a- measure be removed by a judicious application y m a l5 S_ of the impelling power, and by proportioning clous appii- the size of the friction wheels to the pressure c f "- n of , .... _ r the mipel- which they severally sustain. It we suppose, ling power. for example, that the weight of a wheel, whose iron gudgeons move in bushes of brass, is ICO pounds ; then the friction arising from both its gudgeons will be equivalent to 25 pounds. If we suppose also that a force equal to 40 pounds is employed to impel the wheel, and acts in the direction of gravity, as in the case of overshot wheels, the pressure of the gudgeons upon their supports will thus be 14O pounds and the fric- tion 35 pounds. But if the force of 40 pounds could be applied in such a manner as to act in direct opposition to the wheel's weight, the pressure of the gudgeons upon their supports would be 10O 40, or 60 pounds, and the fric- tion only 15 pounds. It is impossible indeed to make the moving force act in direct opposition to the gravity of the wheel, in the case of water- mills ; and it is often impracticable for the en- gineer to apply the impelling power but in a given way ; but there are many cases in which the moving force may be so exerted, as at least in such a manner to increase the friction which arises from the wheel's weight. though this is by HO means a sufficient reason for their adoption, yet we humbly conceive that the errors of the learned should always be opposed with respectful diffidence. We are of opinion, however, and we presume that every person who understands the subject will agree with us, that friction wheels, if properly executed, are of immense ser- vice, and that nothing but the ignorance or narrowness of the proprietors of n achinery could have prevented them from being more generally adopted. 35O MECHANICS. When the moving force is not exerted in a perpendicular direction, but obliquely as in un- dershot wheels, the gudgeon will press with greater force on one part of the socket than on any other part. This point will evidently be on the side of the bush opposite to that where the power is applied, and its distance from the low- est point of the socket, which is supposed cir- cular and concentric with the gudgeon, being H called x we will have Tang, x ~ -p., that is, the tangent of the arch contained between the point of greatest pressure and the lowest point of the bush, is equal to the sum of ail the horizontal forces, divided by the sum of all the vertical forces and the weight of the wheel, H represent- ing the former, and ^"the latter quantities. The point of greatest pressure being thus determined, the gudgeon must be supported at that part by the largest friction wheel, in order to equalize the friction upon their axles. The application of these general principles to particular cases is so simple as not to require air,; illustration. To aid the conceptions, how- ever ., of the practical mechanic, we may mention two cases in which friction wheels have been successfully employed. Mr. Gottlieb, the constructor of a new crane, has received a patent for what he calls an anti- attrition axle-tree, the beneficial effects of which he has ascertained by a variety of trials. It con- sists of a steel roller about 4 or 6 inches long, which turns within a groove cut in the inferior part of the axle. When wheel-carriages are at rest, Mr. Gottlieb has given the friction wheel its proper position; but it is evident that the point of greatest pressure will change when they MECHANICS. 351 are put in motion, and will be neater the front of the carriage. This point, however, will vary with the weight of the load ; but it is sufficient- ly obvious that the friction roller should be at a little distance from the lowest point of the axle- tree. Mr. Gammett of Bristol has applied friction rollers in a different manner, which does not, like the preceding method, weaken the axle-tree. Instead of fixing them in the iron part of the axle, he leaves a space between the nave and the axis to be filled with equal rollers almost touch- ing each other. The axis of these rollers are inserted in a circular ring at each end of the nave, and these rings, and consequently the rollers, are kept separate and parallel, by means of small bolts passing between the rollers from one side of the nave to the other. In wheel-carriages constructed in the common manner with conical rims, there is a great degree of resistance occasioned by the friction of the linch pins on the external part of the nave, which the ingenious mechanic may easily remove by a judicious application of the preceding principles. As it appears from the experiments of Fer- guson and Coulomb, that the least friction is generated when polished iron moves upon brass, the gudgeons and pivots of wheels, and the axles of friction rollers, should all be made of polish- ed iron, and the bushes in which these gudgeons move, and the friction wheels should be formed of polished brass. 6 When every mechanical contrivance has been Friction adopted for diminishing the obstruction which by 811 ._ gments. M. dc La Hire recommends the sockets or bushes to be made square and not concave. 352 MECHANICS. arises from the attrition of the communicating parts, it may be still farther removed by the judicious application of unguents. The most proper for this purpose are swines grease and tallow, when the surfaces are made of wood, and oil when they are of metal. When the force with which the surfaces are pressed toge- ther is very great, tallow will diminish the fric- tion more than swines grease. When the wood- en surfaces are very small, unguents will lessen their friction a little, but it will be greatly di- minished if wood moves upon metal greased with tallow. If the velocities, however, are increased, or the unguent not often enough renewed, in both these cases, but particularly in the last, the unguent will be more injurious than useful. The best mode of applying it, is to cover the rubbing surfaces with as thin a stratum as pos- sible, for the friction will then be a constant quantity, and will not be increased by an aug- mentation of velocity. By the in sm all works of wood, the interposition of tne powder of black lead has been found very useful in relieving the motion. The ropes of pulleys should be rubbed with tallow, and when- ever the screw is used, the square threads should be preferred. 353 MECHANICS. ON THE NATURE AND OPERATION OF FLY WHEELS. FLY in mechanics is a heavy wheel or cy- Fly wheels linder which moves rapidly upon its axis, and is applied to machines for the purpose of render- ing uniform a desultory or reciprocating motion, arising either from the nature of the machinery, from an inequality in the resistance to be over- come, or from an irregular application of the impelling power. When the first mover is in- Causesof * . . j incqual mo- animate, as wind, water, and steam, an mequa- ti0 ninma- lity of force obviously arises from a variation chines. in the velocity of the wind, from an in- crease of water occasioned by sudden rains, or from an augmentation or diminution of the steam in the boiler, produced by a variation in the heat of the furnace ; and accordingly va- rious methods have been adopted for regulating the action of these variable powers. The same inequality of force obtains when machines are moved by horses or men. Every animal exerts its greatest strength when first set to work. Af- ter pulling for some time its strength will be im- paired, and when the resistance is great, it will take frequent, though short relaxation, and then commence its labour with renovated vigour. Pol. II. Z 354 MECHANICS. These intervals of rest and vigorous exertion must always produce a variation in the velocity of the machine, which ought particularly to be avoided, as being detrimental to the communi- cating parts as well as the performance of the machine, and injurious to the animal which is These in- employed to drive it. But if a fly, consisting equalities either of cross bars, or a massy circular rim, be byTfly. connected with the machinery, all these incon- veniencies will be removed. As every fly wheel must revolve with great rapidity, the momentum of its circumference must be very considerable, and will consequently resist every attempt either to accelerate or retard its motion. When the machine, therefore, has been put in motion, the fly wheel will be whirling with an uniform cele- rity, and with a force capable of continuing that celerity when there is any relaxation in the im- pelling power. After a short rest the animal re- news his efforts, but the machine is now moving with its former velocity, and these fresh efforts will have a tendency to increase the velocity: the fly, however, now acts as a resisting power, re- ceives the greatest part of the superfluous mo- tion, and eauses the machinery to preserve its original celerity. In this way the fly secures to the engine an uniform motion, whether the animal takes occasional relaxation or exerts his force with redoubled ardour. Exempli- We havL> already observed, that a desultory flashing or var iable motion frequently arises from an in- machine, equality in the resistance, or work to be perform- ed. This is particularly manifest in thrashing- mills, on a small scale, which are driven by water. When the corn is laid inequally on the feeding board, so that too much is taken in by the fluted rollers, this increase of resistance in- BfcECHANICS. 355 slantiy affects the machinery, and communicates a desultory or irregular motion even to the water wheel or first mover. This variation in the velocity of the impelling power may be dis- tinctly perceived by the ear in a calm evening^ when the machine is at work. The best method of correcting these irregularities is to employ a fly wheel, which will regulate the motion of the machine, when the resistance is either augment- ed or diminished. In machines built upon a large scale there is no necessity for the interpo- sition of a fly, as the inertia of the machinery supplies its place, and resists every change of motion that may be generated by an inequal ad- mission of the corn. A variation in the velocity of engines arises irreguiari- also from the nature of the machinery. Let us^ 8 ^"^ suppose that a weight of 10OO pounds is to be nature of raised from the bottom of a well 50 feet deep, the machl ~ by means of a bucket attached to an iron chain which winds round a barrel or cylinder ; and that every foot in length of this chain weighs 2 pounds : it is evident that the resistance to be overcome in the first moment is 1000 pounds, added to 50 pounds, the weight of the chain ; and that this resistance diminishes gradually, as the chain coils round the cylinder, till it becomes only 1OOO pounds, when the chain is completely wound up. The resistance therefore decreases from 105O to 10OO pounds ; and, if the impel- ling power is inanimate, the velocity of the buck- et will gradually increase ; but if an animal is employed, it will generally proportion its action to the resisting load, and must therefore pull with a greater or less force, according as the bucket is near the bottom or top of the well. In this case, however, the assistance of a fly may be Z2 356 MECHANICS, dispensed with, because the resistance diminish- es uniformly, and may be rendered constant, by making the barrel conical, so that the chain may wind upon the part nearest the vertex at the commencement of the motion, the diameter of the barrel gradually increasing as the weight diminishes. In this way the variable resistance will be equalized much better than by the ap- plication of a fly-wheel ; for the fly, having no power of its own, must necessarily waste the im- pelling power. Having thus pointed out the chief causes of a variation in the velocity of machines, and the method of rendering it uniform by the invention of fly-wheels, the utility, and, in some instances, the necessity of this piece of mechanism, may be more obviously illustrated by shewing the pro- priety of its application in particular cases. Advanta- In the description which has been given of wheels^- Vauloue's pile engine, 1 the reader must have empiified remarked a striking instance of the utility of fly- m the pile wnee t St The ram Q j s raised between the engine. . r i PLATE ix, guides b b, by means or horses acting against Fg. i. t ne levers S S ; but as soon as the ram is elevated to the top of the guides, and discharged from the follower G, the resistance against which the horses have been exerting their force, is sudden- ly removed, and they would instantaneously tumble down, were it not for the fly 0. This fly is connected with the drum B, by means of the trundle X: and as it is moving with a very great force, it opposes a sufficient resistance to the action of the horses, till the ram. is again taken up by the follower. See Vol. i, p. 118. MECHANICS. 35*7 When machinery is driven by a single stroke in the *in- steam engine, there is such an inequality in the gle strokc 11- tr j j steam en- impelling power, that, for 2 or 3 seconds, it doesginc. not act at all. During this interval of inactivity, the machinery would necessarily stop, were it not impelled by a massy fly-wheel of a great dia- meter, revolving with rapidity, till the moving power again resumes its energy. If the moving power is a man acting with a in the com. handle or winch, it is subject to great inequali- monwinch ' ties. The greatest force is exerted when the man pulls the handle upwards from the height of his knee, and he acts with the least force when the handle, being in a vertical position, is thrust from him in a horizontal direction. The force is again increased when the handle is pushed downwards by the man's weight, and it is diminished, when the handle, being at its low- est point, is pulled towards him horizontally. But when a fly is properly connected with the machinery, these irregular exertions are equal- ized, the velocity becomes uniform, and the load is raised with an equable and steady mo- tion. In many cases, where the impelling force is alternately augmented or diminished, the per- formance of the machine may be increased by rendering the resistance unequal, and accom- modating it to the inequalities of the moving power. Dr. Robison observes, that ' there are * some beautiful specimens of this kind of ad- * justment in the mechanism of animal bodies.' Besides the utility of fly-wheels or regulators Fly-wheels of machinery, they have been employed for ac- * cc " r T~ cumulating or collecting power. If motion is 3 communicated to a fly-v,heel by means of a small force, and if this force is continued till the Z3 358 MECHANICS. wheel has acquired a great velocity, such a quarir tity of motion will be accumulated in its circum- ference as to overcome resistances, and produce effects, which could never have been accomplish- ed by the original force. So great is this accu- mulation of power, that a force equivalent to 2O pounds, applied for the space of 37 seconds to the circumference of a cylinder, 20 feet diame- ter, which weighs 4713 pounds, would, at the distance of one foot from the centre, give an im- pulse to a musket ball equal to what it receives from a full charge of gunpowder. In the space of 6 minutes and 10 seconds, the same effect would be produced, if the cylinder was driven by a man who constantly exerted a force of 2O pounds at a winch 1 foot long. 2 Exempli- This accumulation of power is finely exem- fied m the p}j^ e( j j n fa e s i m p-. When the thoner which con- r i . iiijri tains the stone is swung round the head or the slinger, the force of the hand is continually ac- cumulating in the revolving stone, till it is dis- charged with a degree of rapidity which it could never have received from the force of the hand alone. When a stone is projected from the hand itself, there is even then a certain degree of force accumulated, though the stone only moves through the arch of a circle. If we fix the stone in an opening at the extremity of a piece of wood 2 feet long, and discharge it in the usual way, there will be more force accumulated than with the hand alone, for the stone describes a larger arch in the same time, and must therefore be projected with greater force. 2 This has been demonstrated by Mr. Atwood. See his Treatise on Rectilineal and Rotatory Motion. 3 MECHANICS. 359 When coins or medals are struck, a very con- siderable accumulation of power is necessary, and this is effected by means of a fly. The force is first accumulated in weights fixed in the end of the fly; this force is communicated to two levers, by which it is farther condensed : and from these levers it is transmitted to a screw by which it suffers a second condensation. The stamp is then impressed on the coin or medal by means of this force, which was first accumulated by the fly, and afterwards augmented by the intervention of two mechanical powers. 3 Notwithstanding the great advantages of fly- importance wheels, both as regulators of machines, and col- ^/jj 10 " 8 lectors of power, their utility wholly depends wheels pro- upon the position which is assigned them, rela-P erl y- tive to the impelled and working points of the engine. For this purpose no particular rules can be laid down, as their position depends al- together on the nature of the machinery. We may observe, however, in general, that when fly-wheels are employed to regulate machinery, they should be near the impelling power ; and, when used to accumulate force in the working point, they should not be far distant from it. In hand-mills for grinding corn, the fly is for the most part very injudiciously fixed on the axis to which the winch is attached ; whereas, it should always be fastened to the upper mill-stone, so as to revolve with the same rapidity. In the first position, indeed, it must equalize the varying ef- J In the article on the Steam Engine, the reader will see an account of a new kind of fly, called the conical pendu- lum, which Messrs. Watt and Boulton have very ingeni- ously employed for regulating the admission of steam into the cylinder. 360 MECHANICS. forts of the power which moves the winch j but when it is attached to the turning mill-stone, it not only does this, but contributes very effectu- ally to the grinding of the corn. Dr. Desaguliers mentions an instance of a blundering engineer, who applied a fly-wheel to the slowest mover of the machine, instead of the swiftest. The machine was driven by 4 men, and when the fly was taken away, one man was sufficiently able to work it. The error of the workman arose from his conceiving, like many others, that the fly added power to the machine ; but we presume, that Dr. Desaguliers himself has been accessory to this general misconception of its nature, by denominating it a mechanical power. 4 By the interposition of a fly, however, as the doctor well knew, we gain no mechanical force ; the impelling power, on the contrary, is wasted, and the fly itself even loses some of the force which it receives, by the resistance of the air. 4 Dr. Desaguliers calls it a mechanical organ ; but he gives the same appellation to the lever, and all the other mechanical powers. See his experimental Philosophy, Vol. i, p. 344. 361 MECHANICS. ON" THE CONSTRUCTION AND EFFECTS OF MA- CHINES,' By Mr. John Leslie, Professor of Mathematics in the Uni- versity of Edinburgh. 1. IT is a principle in statics, that, if a body act upon another by the intervention of machinery, an equilibrium will obtain when their potential velocities are reciprocally as their masses. If the power exerted be augmented beyond what is barely sufficient to maintain the balance, a motion will immediately commence, and if it be still increased, the velocity will continually 1 This excellent paper, which Professor Leslie was so kind as to communicate to the editor, was written at Lon- don so early as February 1790. The same subject was af- terwards (in 18O1) treated at great length by the late Dr. Robison, in the art. Machinery, Sup. Encycl. Britan. ; and we do not conceive that we are derogating in the least from the talents of that learned and good man, when we say, that the present paper is written with greater perspicuity, and gives a more elementary and connected view of this inter- esting subject. At some future period Mr. Leslie intends tQ resume the investigation of this subject. En. 362 MECHANICS. increase. But this velocity will increase in a smaller ratio than the power; and there will, therefore, be a certain point of augmentation, at which the force employed will produce the great- est proportional effect. Such is the grand object that we ought to have always in view in the con- struction of machines. 2. Forces have been divided into two kinds ; those \vhose action is supposed to be instantan- eous, and those whose action is continued and in- cessant. The former have been termed impuls- ive, the latter accelerating or retarding. Though accelerated or retarded motions perpetually oc- cur to our observation, the ancients seem to have admitted no other force but that of im- pulsion. It is difficult, indeed, to conceive, that a body can act at a distance ; and the idea that motion is always communicated by contact, is one of our earliest and strongest prejudices. Sir Isaac Newton himself was in this instance carried away by the current of opinion. His theory of aether was an attempt to explain gra- vitation by impulsive forces. But there are many facts and experiments which satisfactorily prove, that between the particles of matter there subsists a repulsion, increasing as the distance diminishes, and that no absolute contact cm ever take place. A body does not acquire its cele- rity in an instant. Nothing material can exist but what infinite ; and the beautiful law of con- tinuation, by which changes are produced by im- perceptible shades, can never be violated. But an amazing force may be exerted, and an effect may be produced, in a time so small as to elude the acuteness of our senses. Hence the origin of our idea that motion is derived from impulse. If, however, we consider the subject with more MECHANICS. attention, \ve shall find that it is really as diffi- cult to conceive action in contiguity as at a dis- tance. In neither case can we deduce the con- sequences a priori. The connexion which sub- sists between cause and effect is not necessary and absolute ; it is founded upon the invariable experience of our senses. We may, therefore, conclude, that there is only one kind of force, and that is the accelerating or the retarding. Hence it will always be possible to determine the proportional intensity of any given force, com- pared with that of gravity, and to assign a weight, which, by its pressure alone, would in a given time produce the same effect. 3. If the gravity of an elementary point at the earth's surface be denoted by 1, the whole attractive force will be as the number of points, or as M, the mass of the body. Let F express the intensity of another force urging the same body j then My^F will denote the quantity of force exerted, or * ; but -^y jF; wherefore, the intensity of a force is directly as its quantity, und inversely as the mass which is urged. 4. Let y~~ the velocity of a body, S~ the space described, and T the time of descrip- tion ; the velocity that is acquired may be conceived to be composed of all the successive augmentations which are produced by the con- tinued exertion of the force, and which are pro- portional to the intensity of its action. But the force may for a moment be conceived to be uniform ; whence the increment of velocity is compounded of the force, and of the increment of the time, or P==FT. Suppose the force to be constant, then i^=FT ; and when the time is j * MECHANIC*. given, the velocity must be as the accelerating force. Let F=CFT, and Fand T denote feet and seconds of time. When -f 1, and T~l, we shall have /^zz C the velocity acquired by descent at the surface of the earth at the end of the first second. Put dz=. 16.1 feet, then C=2, j . . , _ - - - and reducing, v*pz~pvw zvw w , and transposing, v*pz vwzzzpvw w* } whence pww iv* z= v ip +vv/ '9 consequently the real action upon the weight, or vzzz 366 MECHANICS. 8. Hence the intensity of the force which , . , ,-, . 1 fp'U'W - W* \ pV - IV urges the weight, or b is [-, , - jor-^-r - ; w \ v 2 p + ^-1U p-\-1U effect increases more slowly than the power. Thus, according as the power is equal to the weight, is double, triple, or quadruple, &c. the intensity of action is O, T , i, -J-, , &c ..... and ultimately zrl. The comparative intensity is, therefore, O, J-, i, ^, T V, y T , -^ &c. and ulti- mately zzO. If p w 9 the intensity of action is I V - 1 \ o 1 J or w (-^r . Suppose the advantage z -- WO - IV I V - 1 to be successively 1, 2, 3, 4, 5, &c. then the intensity of action is O, f, y, T 3 -, T y 9 -/y, ^7, &c. and ultimately vanishing. Both these series commence at zero, increase, become stationary, and then continually decrease, till they vanish. In the present case, the maximum must lie be- tween the 2 d and 3d terms ; for these are equal in both. 10. Since the intensity of action produced 11 pi) - W i . . by the power p is r , the comparative inten- sity, or the effect produced by a given force 1, MECHANICS. 367 11 i_ 1 F *" P V W T-l.' will be rr Xr-~i or + * * , * This quan- p PV-+-ZU p v -\-piu tity is, therefore, the proper measure of effect, and to increase it must be our great object in improving the machine Let the power and weight be constant ; to find the value of v, when the comparative intensity of action is a maxi- mum. By taking the fluxion of fT"*? -.we - J- J obtain ^- v *"+p w )* =O; whence, by transposition and reduction, p 3 v* -f-j&'wrz transposing, /w* 2w;*; w, and dividing, v z V=L , and resolving the quadratic, r zi P P b P W \ i IV ~F> v =j -\-ivp) TT T i i ^ f . Hence, according as the weight is equal to the power, is double, triple, or qua- druple, &c. the advantage ought to be 1 -j-^/2, , 3-{Vl 2, 4-fV20, 5+^/30, 6+^42, 1 1 . When w is very large compared with p J' 1 -' < * , . ft - . , the expression ^ - ! * is nearly . In most cases, it will be sufficiently accurate to suppose v = ; and hence, in order that a ma- chine may produce the greatest possible effect, without increasing the power applied, the advan- tage which would procure an equilibrium ought to be at least doubled. Substituting this value in the formulas in art. 9, we obtain 1> e 2dT -, and S=dT> 12. If tbc true value of v , or 368 MECHANIC^. be substituted in those formulas, we shall ob- tain ^rfc^^^v.-d * when the power is equal to the weight, the * 4/2 A/2 i greatest intensity is - ^ , 2 or *^j , or about one fifth of the force of gravity. If w be sup- posed to be successively ^:2p', 3p, 4&, c. the e ' -11 u A/6 A/12 intensity of action will be ya+18via . ** tO 3QO + 50V/3' __, __, _L. ? &c. derived from the expressions in the last article. If the weight be great in re- spect of the power, the intensity of action will , Hence the other formulae will be found ; ^za 2dT f-!&*L\ and S = * Wherefore, in a machine constructed in the best manner, the accelerating force which impels the weight never amounts to one fourth of the gra- vity of tne power. 13. Let the weigh* and advantage be giv.en, and let it be required to find the power, when the measure of effect or comparative Intensity of action is a maximum. Suppose p to be variable in the expression A z ~,^ of art. p V -}-pw 10-, and taking the fluxion, we shall have } Q h -\-w) (pv w}> and reducing* v 3 P * 4" VW P =: 2 v 3 p z + pvw 2i> * wp w * , and transposing, v 3 p*2v*wpzziv j ' and dividing* MECHANICS. p* -- - p-=*~- and resolving the quadratic, ? . tv 9 P -- =\/(-T -f-T ) and P -- Hv/f 4- T v \v* ' V / T> ' v l rT' * *o* Hence if the advantage be 1, 2, 3, 4, .5, &c. the power ought to be w (l-f-^/2), w (i+y/^j 14. If v be large, the value ofp will be near- 2w . w 2tt)f . . 1 \ . , ' '2w iy = -- {-;r-r> or -~l 1+ 1, or m general - J v * 2v */ But when v is large, these formulae will. t>e v ex>- pressed with sufficient accuracy thus, FzZldT wdS=dr>(). UW rtot) -, , . Let the true value or Z> or 4-v/ x U +. fj -i? %=l, and v be denoted by 1,2, 3, 4, 5, &c. the corresponding absolute intensity will be % ?\/ 2 > or \/ 2 1? y/20 .y/30 ' 204-^/320' 17. If the accurate value of p be substituted in the expression *jrrr- f r l he compamtive in- u* l f v/(v a 4-v tensity, we obtain f _ . . T? - 7 u> V2t> + 2 -f 2 >/ x Suppose wir 7 and v successively =r 1 , 2, 3, 4, 5, &c. then the measure of the effect will be V 2 or q /fi ~ V ' __ _ 244. ^432 -f v'l 2>40 -f V 1 2804- /W?Q+ -V/30OO+ v^25 &c. Let v be a large number, the proxi- mate measure of the effect will then be z: - , ,. or ; and thls expres " sion will be ultimately rrf. Wherefore, com- paring this result with that in art. 1 2, it appears, that, in whatever way the maximum be procur- ed, the force which impels the weight can never amount to one fourth part of the direct action of the power, 1 8. Hitherto we have not taken into account the force expended in impressing motion upon the parts of the machine which connect the power and weight. Let a, b, c, d, &c. denote the masses of the communicating parts, and let , , y, &, &c be proportional to their corre- sponding velocities, and Q to that of the weight, MECHANICS. 371 It is obvious, that the momentum of the part a is equal to the momentum of the mass a ~ : In the same manner, the momentum of b, c, d, &c. will be equal to that of the quantities of matter -j, ^-, , &c. moving with the celerity of the weight to be raised. But from the permanency of the construction^ these quan- tities are constant. Whence the total quanti- ty of motion is the same with that of a mass ^ Q y , which is given. Let it be equal to q, and supposing, as before, the power =z/ and the weight =r?#, the whole mass, on which the celerity of the weight to be raised must be impressed, will be denoted by w+q. 19. It is obvious that i s still that part of the power which is sufficient to maintain the equilibrium, and that the motion is produced i _ . . IV v-Ty . . . by the remaining part p or . This accelerating force may be resolved into ,r, which is exerted against the compound weight w+ q t and ? v ~~ w X9 or^ 1 ^ which acts directly upon the power p* But the velocity gener- ated in a given time is (5) as the intensity of the force divided by the mass. The velo- city of the power, therefore, will be denoted I .. I / T , ..!_ by-r(^ lor*- . But the exer- / j> \ v ) v tion of the force x, which urges the compound weight, is, in consequence of the mechanism^ equal to the direct action of vx. Whence the celerity acquired in the same time will be ex- Aa2 372 MECHANIC. pressed by v * . Therefore, from the condi- J> - , . vx pv Vx iu tions of the motion ; : * : : 1 : v; con- sequently ^==^~^"" W > and reducing, qvx X wvx rr pvw w z -f- pqv qw 9 and divid- ing x pvww'+pqv qw ^ (pv)( } " pv" \-qv-\-ivv ' pv 3 4. qp Whence vx 9 the real force exerted upon the j u^ (P v wj^iv + q) rrn. compound weight, is =t ^ +g+ w Th e m- tensity of force is, therefore, --z= r^~ W . 7 /y -f-y-f 10 Hence we shall obtain expressions for the space described, and the velocity of description. For '20. Since the intensity of action is zz measure o f effect will be rr Supposing 6 to be variable, and taking the fluxion, we shall obtain, when the effect is a maximum, Whence p*v*-}-pqv -\-pwv n (2pv* -\-q-\-w) (pv w) ~ t 2p*v* -{- Pq v ~\~ P wv tyv'tv qw w*i and transposing, p*v* Q.pv*iv .qw "i T i 2,w nw-^-iv 1 '^'i' ^Htf-.a and dividing, p* *P~- i ? anq.re- o r v * r ^,3 solving the quadratic, j& ~-f-\/("~i~i~^T ~f~^0 If v be large, the value of p will be nearly O.yt .. n ~4p~fciJ i " Whence in machines, where the advantage is great, we may disregard the mo* MECHANICS. 373 menta of the connecting parts, and consider the force which ought to be employed as double of what is barely able to maintain the equili- brium. 21. In our investigations, we have always supposed that the same accelerating force is uni- formly exerted. But instances frequently occur, where the power applied increases or diminishes during the action of the machine. This varia- tion may be affected by numberless circum- stances, and the general hypothetical solution of the problem would involve tedious and com- plicated formulae. We shall content ourselves with a familiar example. Suppose that a weight P is attached to one of the extremities of a rope, JVAEP^ of equal and uniform texture, and ap- PLAT* ix : plied to the circumference of a wheel, and to Flg> 4% the other extremity a smaller weight W is ap- pended. 3 It is manifest, that P will at first descend solely by its excess of weight ; but its exer- tion will be continually increased, from the addition of the portion of the rope BP^ while the antagonist power W suffers an equal dimi- nution. J A machine, constructed upon this principle, is actual- ly employed in some coal works. P is a light capacious bucket, W another that is strong and massy. When both are empty, W descends and elevates P ; W\* then loaded with coals, and, at the same time, a cock is opened which fills P with water. Pthen descends, by its superior weight, and raises the load. But when it reaches the bottom of the pit, it pushes up a valve, the water is discharged, and the action of the machine it renewed. Aa3 374 MECHANICS. f 22. It 'will be proper to take into account the momentum of the wheel. Let it be supposed to be solid and homogeneous, and let the radius of the whole .^Czrr, and of the variable circle CDz=,r, and let *:=: PLATE ix, 6.2832 ; then the minute annulus DEde is Fl "g-3- equal to the rectangle of its length and its breadth, or *xx ; but the velocity is directly as the distance from the centre of motion ; whence the momentum of the annulus will be equal to that of ^-^ applied at A. And since momentum O f 3r ' matter of the wheel is equal to the momentum r r Tf' *"* 1. r. of a quantity of matter or , having the wr* velocity of the point A. But -r-~=: area or _j* A and =:4-^> m& hence the momentum of o the wheel will be the same, if ~ of its matter were collected and accumulated at its circumfer- ence. 23. Now let us denote the weight to be raised, together with that of the rope and -|- of the wheel, jbz: the power applied, Air the length of the rope, or the whole height of ascent, am the weight of the rope, and ^ of the weight of the wheel, and x~ the space through which the ascent is already made. The force applied is therefore =/>-f ? and the re- sistance opposed zzw - ^-, consequently, the accelerating force, which must e the di? MECHANICS. 375 ferencc of these, is =p+b Hh > or But t his must bedifFused through h . the whole mass w+p ; wherefore the intensity r s __ T of action is sr /,&.i. w 6 - ll 3 . ,, ,v , phx + bhx uthx+laxx In art. 5, that VV- 2c*X< * kw h ~ J i TS* oJ ->** and integrating, |r*=:2(f X rr / j y /"/** + x w * x 4- fljc A ; consequently r=2 v /^ v /( <{: - T^H^ -- ) /JL /fP"\~^ it y*i"' J \ hence the final velocity ls=2^/dfiy/^ j^ J 9 24-. Let the power applied be equal to the whole weight of the rope, and suppose that no- thing is appended to the other ; then, if the mo- mentum of the wheel be disregarded, the final velocity will be =2 V /(^X v/ b * I j V" V~ 26. To find the fluent of : . , x put f ^f 3t *T~ /I? *Cj and Tzjrrr Vzx-\-z * ; and transposing, rc# Szlrizz s and dividing, tfzr n _ 2z t and taking the fluxion, zz z z Inzz 2z*z T -= ( - or --- But -Zx-{--Li or substituting the value of z -- Una z z, or -- or reducing, zr _ 2z > whence the expression x Inzz 22z ,. ., j , nz z* - .. T.. ^-rz 7 o i "> dl vided by - -, or yrjif^ltx) (n Iz) 1 J n 2z ' -*-. The fluent of -^- is C Hyperbolic 2 2 n 1z * r Logarithm, n 2z To find the correction, suppose 2zrro ; in this case the fluent vanishes, and 6 Hyp. Log. n ; whence the true fluent is Hyp. Log. n Hyp. Log. n 2z, or Hyp. Log;. - . But z=^/(x 1 4-nx) ,r, and substi- 9 n -2z tutino; the fluent, is Hyp. Log. r- ",. a , . Jr 6 ^_2x 2\/(x i -\-nx) To procure a more convenient expression, MECHANICS. 377 multiply the numerator and denominator by n-j-2 > r-f-2 v /^ 1 -f w ^ i tnen we nave ^fyp Log . -f 2*+2yY* +*x^~ or Hyp> Log< "*"., and resuming the value of w, ph+bhwh ph+bbwh * H or by reduction, T= x avuhx When ar:=A, we obtain for the whole time of the performance T ~ y/ ~ when the comparative quantity of action is a imum, may also be determined, but it will be in- volved in a transcendent equation. 27. When the resistance of the parts of a ma- chine is inconsiderable, we perceive from all these investigations the importance of continuing the action. The successive impulses are retained and accumulated, and the performance constant- ly increases. The whole quantity of action, pro- duced by the machine, is not in the simple ra- tio of the time of continuance, but in that of the square. 28. When we attempt to take the resistance of the moving parts of the machine into the account, we have great difficulties to encounter. Friction is affected by numberless circumstances ; by the nature of the substances employed in the 378 MECHANICS. construction ; by their form ; by the degree of polish ; by their velocity, &c. Nor is it pro- bable that its quantity can be derived from ge- neral principles ; it must often be determined from the individual case, and can never be accurately ascertained. Friction may be con- sidered as a continual retarding force. It may therefore be compared with that of gravity, and its effect may be estimated from that of a coun- teracting weight. The mass of the connecting parts, and their friction, both contribute to di- minish the celerity of the motion ; but they pro- duce this retardation in different ways. The momentum which must be impressed upon the connecting parts of the machine requires a great- er diffusion of power, and thus diminishes in some degree its effect. Friction does not alter the general mass, but reduces 'the quantity of accelerating force, and consequently the intens- ity of its action If the quantity of friction were equal and constant, it is obvious, that if the mov- ing power exceed it, the motion would be per- petually accelerated. But this is very far from the fact ; for all the motions with which we are acquainted tend to an uniform celerity, and in a certain time would actually attain it. We may therefore conclude, that in the same machine, the friction increases in a certain ratio with the velocity. The great desideratum in mechanics is to determine the law of progression, and our de- ductions upon this subject must be considered as merely hypothetical. 29. Suppose, as before, wrr the weight to be raised ; q~ the mass, which, if attached to the weight, would have the momentum of the connecting parts ; jbrz the power employed ; v=; the advantage ; and let the friction be equal MECHANICS. 379 4fe 9, some function of the celerity of the weight. Because the moving parts of the machine rer main the same as before, it is manifest that the intensity of action will be proportioned to the quantity of accelerating force ; whence ajj iy p : p 9 or pv w : pv w w$ : : the true force which con- stantly acts upon the weight. Whence, if the quantity v

Tr rr i t_ mV =-- or F =-. If 9 =mr 9 then 30. We may neglect the performance which is made during the first acceleration of motion as inconsiderable, when compared with the whole. The quantity of action will therefore be as f; and if the power affects the friction only by altering the velocity, the comparative action will be denoted by ; whence the per- formance will be a maximum, when pV~Vp t If, as before, 9=mF-L ; the n f^=^V""' > \ m v J * and MECHANICS. nvp = pv w, and transposing, w pv pvn, .,..,. V) /~\ 1U i and dividing, p = j^. Or put -, the power which would barely maintain an equilibrium =*; then/> =rx^. 31. The law of the increase of velocity at first may also be ascertained. For (art. 4), the time corresponding to the velocity V is = r^jpy and in the present case T = h~ x ^r -fg+u. fr As ^ i s a function of ^ the po w r

^woc DESCRIPTION OF A SIMPLE AND POWERFUL CAPSTANE. g^j } J. HIS capstane is represented in Fig. 1, where PI-ATI ix AD is a compound barrel consisting of two linders C, D of different radii. The rope DECof* simpk is fixed at the extremity of the cylinder Z>, and ca P $ane - after passing over the pulley , which is attach- ed to the load by means of the hook F, it is coiled round the other cylinder (7, and fastened at its upper end. AB is the bar by which the compound barrel CD is urged about its axis, so that the rope may coil round the cylinder D 9 while it unwinds itself from the cylinder C. Let us now suppose that the diameter of the part D of the barrel is 2] inches, while the diameter of the part C is only 20, and Jet the pulley R be 2O inches in diameter. It is evident, then, that when the barrel AD is urged round by a pressure exerted at the point #, 63 inches of rope will be gathered upon the cylinder Z>, and 6O inches will be un winded from the cylinder, C by one revolution of the bar AE^ these num- bers representing the circumference of each cy- linder. The quantity of wound rope, there- fore, exceeds the quantity that is unwound by G3 60, or 3 inches, the difference of their re- spective perimeters ; and the half of this quan- titv, or ly inches, will be the space through -which the load or the pulley E moves by one turn of the bar. Bur if a simple capstane of 882 MECHANICS, the same dimensions had been employed, the length of rope coiled round the barrel by one revolution of the bar would have been 6O inches, Method of and the space described by the pulley or load computing to |j e overcome would have been 30 inches. Now, it is a maxim in mechanics, 1 that the power of any engine is universally equal to the velocity of the impelled point divided by the velocity of the working point, or to the velo- city of the power divided by the velocity of the weight, that is, to the velocity of the point B divided by the velocity of the pulley E ; con- sequently if the lever in both capstanes is the same, and the diameter of their barrels equal, the power of the common will be to the power 4 of the improved capstane as 1^- to SO, that is, inversely as the velocity of their weights, and Qf\ the power of the latter will be ~ =: 20, or in ( \ other words, will be equivalent to a 20 fold tackle of pulleys. * If it is wished to double the power of the machine, we have only to cover the cylinder C with lathes a quarter of an inch thick, so that the difference between the radii of each cylinder may be half as little as before ; for the power of the capstane increases as the difference between the radii of the cylinders is diminished. As we increase the power, there- fore, we increase the strength of our machine, while all other engines are proportionably en- feebled by an augmentation of power. Were we, for example, to increase the power of the common See vol. i, pp. 57, 58. 1 In practice it will be found equivalent to a 26 fold tackle of pulleys, as about one third of the powef of a sys- tem of pulleys is destroyed by friction and the bending of the ropes.- MECHANICS. S83 capstane, we must diminish the barrel in the same proportion, supposing the bar or handspike not to admit of being lengthened, and we not only weak- Convert- en its strength, but destroy much of its power ibl< into a t_ V r t a crane. by a greater flexure or bending or the ropes. The reader will perceive that this capstane may be converted into a crane or windlass for raising weights, merely by giving the compound barrel A B * horizontal position, and substitut- ing a winch instead of the bar AB. The su- periority of such a crane above the common one is obvious from what has been said ; but it has this additional advantage, that it allows the weight to stop at any part of its progress, without the aid of a ratchet wheel and cateh, from the two parts of the rope pulling on contrary sides of the barrel. The rope, indeed, which coils round the larger part of the barrel acts with a larger lever, and consequently with greater force than the other ; but as this excess of force is not suf- ficient to overcome the friction of the gudgeons, the weight remains stationary in any part of its path. A crane of this kind was erected in 1 797 at Bor- denton in New Jersey, by Mr. M'Kean, for the purpose of raising logs of wood to the frame of a sawmill,whichwas 10 feet distant from the ground. The diameter of the largest cylinder was 2 feet, and its length 3 feet ; the other cylinder was 1 foot in diameter, and of the same length with the largest. The difference of their circumferences, therefore, was 3 feet, and the log would move through a space of 1 8 inches with 1 turn of the handspike ; and through the required height with only 8 turns. The length of the bar or handspike was 6 feet, which, at the point where the power was applied, described a circle of about 3O feet, so that the power of the crane was as 1 to 20. The length of the rope was only 55 feet, where- 384* MECHANICS. as if the weight had been raised through the same height with a similar power by means of a tackle of pullies, 27O feet of rope must have been employed. In the latter case, however, the rope sustains only ~ of the weight, but in the former it supports one half of the load. In describing a capstane of this kind, Dr. Ro- bison asserts, 3 that when the diameters of the cylinders which compose the double barrel are as 16 to 17, and their circumferences as 48 to 51, the pulley is brought nearer to the cap- stane by about 3 inches for each revolution of the bar. This however, is a mistake, as the pulley is brought only ly inches nearer the axis. This will be evident, if we conceive a quantity of rope equal to the circumference of the larger cylinder to be winded up all at once, and a quantity equal to the circumference of the lesser one to be unwinded all at once. In the present case, 51 inches of rope will be coiled round the larger part of the barrel by one revolution of the capstane bar, and consequently the load would be raised 25-f- feet, the rope being doubled. Let 48 inches of rope be now unwinded from the lesser cylinder, and the load will sink 24 feet ; therefore 25^- 24 ly feet is the whole height or distance through which theweighthasbeen moved. Dr. Robison observes that the capstane now described was invented by an untaught but ingen- ious country tradesman. It appears, however, to be the invention of George Eckhardt, and wise of Mr. Robert M'Kean of Philadelphia, son to the present governor of Pennsylvania. Mr. Gregory observes, that he has seen a figure of this capstance among some Chinese drawings nearly a century old. _ . - J Encyclopedia Britannica Supplement Art, Mach'tn:ry t vol. xx, p. io). 385 MECHANICS. ACCOUNT OF AN IMPROVEMENT ON THE BALANCE. IT must be a matter of great convenience to the improve, experimental philosopher, as well as the practical j^!^ e nt chemist, to have a method of ascertaining the gravities of bodies without the aid of a number of small weights. The following contrivance has occurred to me as likely to answer this purpose, while it has the advantage of facility in its ap- plication, and accuracy in its results. In Figure 6, FA, FB, represent the arms of PIATE a common balance. A micrometer screw D is fitted to the arm FA, in such a manner, that when it is turned round by the nut Z), it neither approaches to, nor recedes from, the centre of motion F. The screw D C works in a female screw in the small weight ?z, and, by revolving in one direction, carries this weight from S to /?, and thus gives the preponderance to the scale G. When the weight n is screwed close to the should- er S, the scales are in equilibrio ; but when it is made to recede from S, this equilibrium is de- stroyed, and the same effect is produced as if an additional weight had been put into the scale G. In most cases, it might be preferable to make the cales in equilibrio, when the weight n is equi- ~l.II. Bb 386 MECHANICS, ^ distant from S and R. The recession of the weight n from the shoulder S, is measured by a scale of equal parts engraven on the arm A F, Each unit of this scale is equivalent to one revo- lution of the screw, and is subdivided into 100 parts by a divided scale on the circumference of the nut D, to which the projecting part of the shoulder S is the index. Let us now see what weight put into the scale G, or suspended at A^ will be equivalent to the given weight n, when moved to a given distance from S. It is evident from the property ,pf the lever, that if x be the equivalent weight Snx n, required, FAiSn'zzn'.x, and x n .-> t A that is, the real weight n exceeds the equiva- lent weight x, as much as FA exceeds Sn. Let n be z= 20 grains, S n = 5 tenths of an inch, 20 X. 5 and F A =. 10 inches, then x =r ~ 1 grain ; so that by shifting a weight 2O grains 5 tenths of an inch from S, the same effect is pro- duced as if 1 grain had been thrown into the scale G. By having several weights instead of , the utility of this contrivance may be mucfi increased. 387 :n '.-t.ji : 'jf-t ;~(o !?f jioq n aiih fa o-rertv/sfcroa r/d Hw. sirAattth 3(f? ?*jj'.^X\i 3i m-aifv/ inioTsrfj :-r 7fiiE>r/p9J?n(Xf MECHANICS, '. ;!KD li 3fi) oliilataS w 31 v^ 4 '^' - f ' ? ^ - : i.iJiod.K :-3i cisrij r b'j-;n^iiid '10/1 bo^n-xrei'- 1 su ^ MECHANICAL METHOD OF FINDING THE CENTRE OF GRAVITY; o - n bifiod odi i;JiJ tOiinr^'a ?; ri^i .: .1 -..-d" o; XA.S it is frequently necessary in mechanical operations to find the centre of gravity, the fol- lowing practical method may probably be accept- able to some readers, as it is not to be met with in any of the elementary treatises in our lan- guage. 1 . If the body, whose centre of gravity is to be found, can be easily suspended by a thread or cord, then the centre of gravity will be situ- ated in some point in the direction of the cord prolonged. Suspend the body at another part, so that the new direction of the cord may be nearly at right angles with its former direction ; then, as the centre of gravity must lie some- where in the new direction of the cord prolong- ed, the point where these two lines (formed by prolonging the cord) intersect each other, will determine the centre of gravity. 2. If the body is of such a size or quality that it cannot be conveniently suspended, place it upon an horizontal edge so that it may be in equilibrio ; the horizontal edge will make a line or mark on the body in the same direction with itself, and the centre of gravity will be in some point in this line. Balance the Bb2 388 MECHANICS. body a second time, so that the line upon the body may be nearly at right angles to the hori- zontal edge, which will make a new line or mark upon the body/, the centre of gravity therefore will be somewhere in this new line, and consequently in the point where it intersects the former line. 3. If the body is so flexible that it can neither be suspended nor balanced, then let a board be balanced, as in case 2 d , and upon it, when ba- lanced, lay the body, whose centre of gravity is to be found, in such a manner that the board may still be in equilibrio ; then the centre of gravity will be in a line opposite to that which is made on the board by the horizontal edge ; and by shifting the position of the board, and again ba- lancing it, a new line will be found, the inter- section of which, with the former line, will de- termine the centre of gravity. 6 ' hi co od) t) ; -nil Jliv/ i9fljo ihr.3 KJ Oxl^: < ,' > <\ .; f srfl II < i ' rx ' 389 :K HYDRAULICS. ON THE STEAM ENGINE.' :.LJ to >' I X HE superiority of inanimate power to the ex- importance ertions of animals in turning machinery has been ofd } csteam universally acknowledged. In the former, the power generally continues its action without the smallest intermission, but frequent and long re* laxations are necessary for restoring the strength and activity of exhausted animals. There are many places, however, where a sufficient quan- tity of water cannot be procured, or where it cannot be employed for the want of proper de- clivities j and there are situations also which are of wind- BiiUOiSE DULL QUAGLIENI'S GRAND Cl mills can LOWER ABBKY STREET. Open every Evening at 715, commencing at 71 a THE POPULAR QUESTION. rom HAVE YOU BEES TO THE To see the Graceful Lady Riders-the Gentlemen L the Wondrous Acrobats the Talented Gymnasts _ x,., co r\( vellqus Tumblers the Pole Sprites the Antipode:*- CaSC OI Musical, Talkative, Flexible Clowns the Wonder forming Horses, Trained Mules, and E :ucated Pon. M O N DAY, September 4th, Last 6 Nichti ot *i- W'.rld's Wonder, HERE CHRISTOFP, Supported by tne whole strength of the Oomp^ n a t great * Comic.Ijiteri^v^-^ydrodyna. mique, 1/85, torn, i, p. lip, 145, under the title La Machine a Feu ; and also by Pony, in his Nouvelle Archi- tecture Hydralique, part 2". In the last of these works, the construction of the. engine is very minutely described) and illustrated by a great number of engraving!. Bb 3 388 MECHANICS. body a second time, so that the line upon the body may be nearly at right angles to the hori- zontal edge, which will make a new line or mark upon the body^; the centre of gravity therefore will be somewhere in this new line, and consequently in the point where it intersects the former line. 3. If the body is so flexible that it can neither be suspended nor balanced, then let a board be balanced, as in case 2 d , and upon it, when ba- lanced, lay the body, whose centre of gravity is to be found, in such a manner that the board may still be in equilibrio ; then the centre of gravity will be in a line opposite to that which is made on the board by the horizontal edge ; and by shifting the position of the board, and again ba- lancing it, a new line will be found, the inter- section of which, with the former line, will de- termine the centre of gravity. wo mchea in height, and is covered with il drive waer m rve the engine eight minutes The dwarf piece of mechanism is designed and made by a olockmanufaoturer at Horaforth. J HYDRAULICS. ON THE STEAM ENGINE. 1 J. HE superiority of inanimate power to the ex- importance ertions of animals in turning machinery has been oft ^ csteam universally acknowledged. In the former, power generally continues its action without the smallest intermission, but frequent and long re* laxations are necessary for restoring the strength and activity of exhausted animals. There are many places, however, where a sufficient quan- tity of water cannot be procured, or where it cannot be employed for the want of proper de- clivities ; and there are situations also which are highly unfavourable for the erection of wind- mills. But even when water and wind mills can be conveniently erected, there is such a varia- tion in the impelling power, arising from acci- dental and unavoidable causes, that sometimes, in the case of water, and often in the case of 1 The theory of the steam engine is given at great length by the Abbe Bossut, in his Traite de Hydrodyna- mique, 1/85, torn, i, p. lip, 145, under the title La Machine a Feu ; and also by Pony, in his Nouvelle Archi- tecture Hydralique, part 2". In the last of these works, the construction of the, engine is very minutely described, and illustrated by a great number of engravingi. Bb 3 390 HYDRAULICS. wind, there is not a sufficient force for putting the machinery in motion. In such circumstances, the discovery of steam as an impelling power may be regarded as a new sera in the progress of the arts. Wherever fire and water can be obtained, we can procure a quantity of steam capable of overcoming the most powerful resist- ance, and free from those accidental variations of power which affect every inanimate agent that has hitherto been employed as the first mover of machines. History of The invention of the steam engine has been the steam . -111 i -r- T i engine. universally ascribed by the Lnglish to the mar- quis of Worcester, and to Papin by the French ; but there can be little doubt that about 34 years prior to the date of the marquis's invention, and about 61 years before the publication of Pa- pin's, steam was applied as the impelling power of a stamping engine by one Brancas an Italian, who published an account of his invention in the year 1 629. It is extremely probable, however, that the marquis of Worcester was unacquaint- ed with the discovery of Brancas, and that the fire engine which he mentions so obscurely in his Century of inventions, was the result of his own ingenuity. z The utility of steam as an impelling power being thus known, the ingenious Captain Savary took advantage of this important discovery, and invented an engine which raised water by the ex- pansion and condensation of steam. Several of Savary's engines were actually erected in Eng- land and in France, but they were never capable * It is said that Gerbert employed steam before the time of the marquis of Worcester, to produce tlte sounds for his automata HYDRAULICS* S9l of raising Water from a depth which exceeded 35 feet. The steam engine received great improve- ments from the hands of Newcomen, Beighton, Blakey, and other ingenious men ; but it was brought to its present state of perfection by the celebrated Mr. Watt of Birmingham, one of the . most accomplished engineers of the present age. Hitherto the steam engine had been employed merely as a hydraulic machine for draining mines or for raising water ; but in consequence of Mr. Watt's improvements, it has for a series of years been employed as the impelling power or first mover of almost every species of machinery. Figure 1 , of Plate X, represents one of Mr. PLATE x , Watt's latest engines. C D is the boiler in Fig. i. which the water is converted into steam by the ^fMr'. ptlC heat of the furnace D. It is sometimes made of watt' copper, but more frequently of iron ; its bottom st " c m ' is concave, and the flame is made to circulate round its sides, and is sometimes conducted by means of flues through the very middle of the wa- ter, so that as great a surface as possible may be exposed to the action of the fire. In some of Mr. Watt's engines, the fire contained in an iron ves* sel was introduced into the very middle of the water, and the outer boiler was formed of wood, as being a slow conductor of heat. When the furnaces are constructed in the most judicious manner, 8 square feet of the boiler's surface must be acted upon by the fire or the flame, in order to convert 1 cubic foot of water into steam in the space of an hour ; and this effect will be produced by between - or -~ of a bucket of good Newcastle coals. When fire is applied to the boiler, the water does not evaporate into steam till it has reached the temperature of 212 HYDRAULICS. of Fahrenheit, or the boiling point. This arises from the superincumbent weight of the atmo- sphere ; for when the water is heated in a vessel exhausted of air, steam is generated even below the temperature of 96, or blood-heat. When the water, however, is pressed by air or steam more condensed than the atmosphere, a temper- ature greater than 212 is necessary for the pro- duction of steam ; but the heat requisite for this purpose increases in a less ratio than the press- ures to be overcome. The steam which is pro- duced in the boiler CD is about 180O times rarer than water, and is conveyed through the steam-pipe CE into the cylinder G, where it acts upon the piston q, and communicates mo- Method of tion to the great beam A B. But before we supplying proceed to consider the manner by which this with waur. niotion is conveyed, we shall point out the very ingenious method which Mr. Watt has employed for supplying the boiler regularly with water, and preserving it at the same height P. This is absolutely necessary in order that the quantity and elasticity of the steam in the boiler may be always the same. The small cistern w, placed above the boiler, is supplied with water from the hot well h by means of the pump z and the pipe f. To the bottom of this cistern is fitted the pipe u r which is immersed in the water P, and is bent at its lower extremity in order to prevent the entrance of the rising steam. A crooked arm u d attached to the side of the cistern u, supports the small lever a b', which moves upon d' as a centre. The extremity b' of this lever carries, by means of the wire b' P, a stone or piece of metal P, which hangs just below the surface of the water in the boiler, and the other extremity a is connected by the wire d u with a valve at HYDRAULICS* 893 the bottom of the cistern u, which covers the top of the pipe u r. Now, it is a maxim in hyd- rostatics," that when a heavy body is suspended in a fluid, the body loses as much of its weight as the quantity of water which it displaces. When the water P, therefore, is diminished by part of it being converted into steam, the up- per surface of the body P will be above the wa- ter, and its weight will consequently be increased in proportion to the quantity of the body that is out of the water; or, to speak more precisely, the additional weight which the body P receives by a diminution of the water in the boiler, is equal to the weight of a quantity of the fluid, whose bulk is the same as the part of the body P which is above the water. By this addition to its weight, the stone P will cause the extremity b' of the lever to descend, and by elevating the arm a d' t will open the valve at the top of the pipe u r, and thus gradually introduce a quantity of water into the boiler, equal to that which was lost by eva- poration. This process is continually going on while the water is converting into steam ; and it is evident that too much water can never be intro- duced ; for as soon as the surface of the water coincides with the upper surface of the body P, it recovers its former weight, and the valve at u shuts the top of the pipe u r. In order to know the exact height of the water in the boiler, two cocks k and / are employed, the first of which, k y reaches to within a little of the height at which the water should stand, and the other, /, reaches a very little below that height. If the water stands at the desired height, the cock k being 5 See vol. i, p. 163. 394 HYDRAULICS; opened will give out steam, and the cock / wii) emit water, in consequence of the pressure of the superincumbent steam on the water P but if water should issue from both cocks, it will be too high in the boiler, and if steam issues from both, it will be too Low. As there would be great danger of the boiler's bursting if the steam should become too strong, it is furnished with Safety the safety valve #, which is loaded in such a 'valve. manner, that its weight, added to that of the at- mosphere, may exceed the pressure of the in- terior steam when of a sufficient strength. As soon as the steam becomes so elastic as to en- danger the boiler, its pressure preponderates over the pressure of the safety valve and the atmo- sphere. The valve therefore opens, and the steam escapes from the boiler, till its strength is suf ficiently diminished, and the safety valve shuts by the predominance of its pressure over that of the interior steam. By opening the safety valve> the engine may be stopt at pleasure. A small rectangular lever, with equal arms, is fixed upon the side of the valve, and connected with its top. To one of these arms a chain is attached, which passes over a pulley from a horizontal to a ver- tical direction, and by pulling which, the safety valve is opened, and the machine stopped. From the dome of the boiler proceeds the steam-pipe CE y which conveys the steam into the top of the cylinder G by means of the steam- valve a, and into the bottom of the cylinder by means of the valve c. The branch of the pipe which extends from a to c is cut off in Fig. 1, in order to shew the valve b, but is distinctly rt s- * visible in Fig. 2, which is a view of the pipes and valves in the direction FM. The cylinder G is sometimes inclosed in a wooden case, in HYDRAULICS. 395 order to prevent it from being cooled by the Construe ambient air ; and sometimes in a metallic case, that it may be surrounded and kept warm by a quantity of steam, which is brought from the steam-pipe E C, through the pipe E G, by turn- ing a cock. It is generally thought, however, that little benefit is obtained by encircling the cylinder with steam, as the quantity thus lost is almost equal to what is destroyed by the cold- ness of the cylinder. After the steam, which was admitted above the piston q by the valve , and is put in motion, by the rope o o which passes over the pulleys o, o, and round the axis o of the fly. Since the velocity of the fly and sun wheel increases and diminishes with the quantity of steam that is ad- mitted into the cylinder, let us suppose that too much is admitted, then the velocity of the fly increase, but the velocity of the vertical 4OO HYDRAULICS. i axis op will also increase, and the balls mn will recede from the axis by the augmentation of their centrifugal force. By this recess of the balls, the extremity p of the lever p s, moving upon y as a centre, is depressed; its other extre- mity s rises, and by forcing the cock at a to close a little, diminishes the supply of steam. The impelling power being thus diminished, the velocity of the fly and the axis op decrease in proportion, and the balls ?;?, n, resume their for- mer position.ono Construe- In Mr. Watt's improved engine, the steam Waives. IC an d eduction valves are all puppet clacks. One PLATE xi, o f these valves, and the method of opening and shutting it, is represented in Fig. 1 of Plate XI. Let it be one of the eduction valves, and let AA be part of the pipe which conducts the steam in- to the cylinder, and MM the superior part of the pipe which leads to the condenser. At OO, the seat of the valve, a metallic ring, of which n n is a section, is fitted accurately into the top of the pipe MM 9 and is conical on the outer edge, so as to suit the conical part of the pipe. These two pieces are ground together with eme- ry, and adhere very firmly when the contiguous surfaces are oxidated or rusted. The clack is a circular brass plate w, with a conical edge ground into the inner edge of the ring n n, so as to be air- tight, and is furnished with a cylin- drical tail in P, which can rise or fall in the ca- vity of the cross-bar AW. To the top of the valve 7, a small metallic rack m F is firmly fast- ened, which can be raised or depressed by the portion E of a toothed wheel, moveable upon the centre D. The small circle D represents a section of an iron cylindrical axis, whose pivots move in holes in the opposite sides of the pipe HYDRAULICS. 401 A A. Its pivots are fitted into their sockets, so as to be air-tight ; and the admission of air is forther prevented by screwing on the outside of the holes necks of leather soaked in rosin or melted tallow. One end of this axis reaches a good way with- out the pipe A A, and carries a handle or spanner b N, which may be seen in Fig. 1, Plate X, and p . LATE x > which is actuated by the plugs 1 , 2, of the rod ' g ' x TN. When the plug 2, therefore, elevates the extremity of the spanner Nb, during the ascent of ifhe piston rod TN, the axle A Plate XI, * A " XI ' Fig. 1, is put in motion, the valve m is raised by means of the toothed racks E and jP, and the steam rushes through the cavity of the circular ring n ??, by the sides of the cross piece of rrfe- tal 00, NN. When the valve needs repair, the cover B, which is fastened to the top of thi valve box by means of screws, can easily be r$ moved. Having thus described the different parts of Mode of the most improved steam engine, 'we 1 shall P eratlon - now attend to the mode of its operation. pUei us suppose that the piston is at the top of the cylinder, as is represented in the figure, and that the upper steam valve a, and the lower eduction or condensing valve d, are opened by means of 'JjJ the spanner JV/, while the lower steam valve c, and the upper eduction valve Z', are shut ; then the steam in the boiler will issue through .the steam pipe CE, and the valve 1 , into the top of the cylinder, and by its elasticity depress the pis- ton, to the very bottom. But when the piston g is brought to the bottom of the cylinder, the extremity B of the great beam is dragged down by the parallel joint TV. Its other extremity A rises, and the wheel U having passed over -j- -of the circumference of $, will have urged forward Vol. II. C 402 HYDRAULICS. the fly-wheel F, and consequently, the machinery attached to it, one complete revolution. When the piston q has reached the bottom of the cy- linder, the piston-ro4 TN of the air pump, by the pressure of the pjug 1 upon the spanner M 9 has shut the steam-valve a, and the eduction- valve d, while the plug 2 has, by means of the spanner .W, opened the eduction-valve , and the steam-valve c. Th steam, therefore, which is above the piston, rushes through the eduction- valve b into the condenser z, where it is convert- ed into water by the jet in the middle of it, and by the coldness arising from the surrounding fluid w, while, at the. same time, a new quantity of steam from the boiler issues through the open steam-valve c, into the cylinder, forces up the piston, and, by raising one end- of the working beam, and depressing, the other, makes the wheel U describe the other semi-circumference of $", )M ^nd causes the fly and the machinery on its axis to perform another complete revolution* As the plugs 1,2, ascend with the piston q, they open or shut the steam and eduction-valves, and the operation of the engine will be thus continued for any length of time. value of ^ From this brief description of the steam-en- Mnprove-" s g me 5 the reader will be enabled to perceive the mem*. nature, and appreciate the value of Mr. Watt's improvements. It had hitherto been the prac- tice to condense the steam in the cylinder itself, by the injection of cold water ; but the water which is injected acquires a considerable degree of heat from the cylinder, and being placed in air, highly rarified, part of it is converted 'into steam, which resists the piston, and diminishes the power of the engine. When the steam is next admitted, part of it is converted into water HYDRAULICS. . 403 by coming in contact with the cylinder, which is of a lower temperature than the steam, in con- sequence of the destruction of its heat by the in- jection-water. By condensing the steam, there- fore, in the cylinder itself, the resistance to the piston is increased by a partial reproduction of this elastic vapour, and the impelling power is diminished by a partial destruction of the steam which is next admitted. Both these inconve- niencies Mr. Watt has in a great measure avoid- ed, by using a condenser separate from the cy- linder, and incircled with cold water ; ' and by surrounding the cylinder with a wooden case, and interposing light wood ashes, in order to prevent its heat from being abstracted by the am- bient air. The greatest of Mr. Watt's improvements consists in his employing the steam both to ele- vate and depress the piston. In the engines of Newcomen and Beighton, the steam was not the impelling power; it was used merely for pro- ducing a vacuum below the piston, which was forced down by the pressure of the atmosphere, and elevated by the counterweight at the other extremity of the great beam. The cylinder, therefore, was exposed to the external air at eve- ry descent of the piston, and a considerable por- tion of its heat being thus abstracted, a corre- sponding quantity of steam was of consequence destroyed. In Mr. Watt's engines, however, 1 Even in Mr. Watt's best engines, a very small quantity of steam remains in the cylinder, having the temperature of the hot well h, or of the water, into which the ejected steam is converted. Its pressure is indicated by a barome- ter, which Mr. Watt has ingeniously applied to his engines lor exhibiting the state of the vacuum. Cc2 404 HYDRAULICS. the external air is excluded by a metal plate at the top of the cylinder, which has a hole it it for admitting the piston-rod ; and the piston it- self is raised and depressed merely by the force of steam. When these improvements are adopted, and the engine constructed in the most perfect man- ner, there is not above ~ part of the steam con- sumed in heating the apparatus ; and, therefore, it is impossible that the engine can be rendered j more powerful than it is at present. It would be very desirable, however, that the force of the piston could be properly communicated to the machinery without the intervention of the great beam. This, indeed, has been attempted by Mr. Watt, who has employed the piston-rod it- self to drive the machinery ; and Mr. Cartwright has, in his engine, converted the perpendicular motion of the piston into a rotatory motion, by means of two cranks fixed to the axis of two equal wheels which work in each other. Not- withstanding the simplicity of these methods, none of them have come into general use, and Mr. Watt still prefers the intervention of the great beam, which is generally made of hard oak, with its heart taken out, in order to pre- vent it from warping. A considerable quantity of power, however, is wasted by dragging, at every stroke of the piston, such a mass of matter from a state of rest to a state of motion, and then from a state of motion to a state of rest. To prevent this loss of power, a light frame of carpentry ' has been employed by several engin- 1 The great beam in Mr. Hornblower's engine, is con- structed in this manner, and is formed upon truly scientific principles. HYDRAULICS. 405 eers, instead of the solid beam ; but after being used for some time, the wood was generally cut by the iron bolts, and the frame itself was often, instantaneously destroyed. In some of the en- gines lately constructed by Mr. Watt, he has formed the great beam of cast iron, and while he has thus added to its durability, he has at the same time diminished its weight, and increased the power of his engine. Encouraged by Mr. Watt's success, several improvements upon the steam engine have been made by Hornblower, Cartwright, Trevethick, and other engineers of this country. But it does not appear that they have either materially in- creased the power of the engine, or diminished its expence. Most of these improvements, on the contrary, excepting those of Hornblower, and the engineers just mentioned, consist merely in having adopted Mr. Watt's discoveries, in such a manner as not to infringe upon his patent. In Figure I of Plate XIV, we have represented PLATE a new form which the steam engine has receiv- Fi g.j jApp ed in France. In particular situations it has some advantages over the common construction; but it is particularly remarkable for the ingeni- Another ous contrivance by which the piston-rod is made form of the to rise in a vertical line. This is effected by steam en ~ means of two beams AB and CZ), moving upon 2 ' the centres B and C, which are always of the same size, though represented otherwise in the figure, for want of room. The sum of the lengths of the two beams, viz. CD+AB, reck- oning from the centres of motion C and , principles. Dr. Robison observes, that it is stronger than a beam of the common form, which contains 20 times its of timber. Cc 3 406 HYDRAULICS. must be equal to the horizontal distance be- tween these centres. The piece of iron A urn has joints at A and m, and is equal in length to the difference of level between the centres of motion C and B. Now, when the beams AB and CD "are in a horizontal posi- tion, the line joining the points ^ and m will form a vertical line ; and when these beams have risen from this position through equal arches, the points A and m will be equally distant from that vertical line, which ought to coincide with the axis of the cylinder produced. The distances of A arid m from this vertical line are obviously the versed sines of the arches which the beams have described ; but as these arches, as well as their radii, are supposed equal, their versed sines, and consequently the distances of A and m from the vertical, are also equal. If, therefore, the piston-rod np be suspended at n, equidistant from A and m, the point n, and consequently the piston-rod, will move in a perpendicular direc- tion. The point, in reality, describes an alge- braic curve ; but when the arches described are small, the deviation from the vertical does not exceed the 10 th of an inch. In this engine / is the cylinder, Q the condenser, P the air-pump, wrought by a chain passing over the arched head, corresponding with op on the other side of CD; K is the pump which supplies the boiler by the pipe KN; LMis the steam-pipe, MG the furnace; His the pump which furnishes the wa- ter in which the air-pump P is immersed, and is wrought by the chain y F and the wheel F. The machine is represented as raising water in the pump R, by means of the chain x E passing over the wheel E ; but when a rotatory motion is re- quired, the beam AB must be prolonged, and men:* HYDRAULICS. 407 the same apparatus fixed at its extremity, as is employed in Watt's steam engine. */ About ten months ago, Mr. Arthur WoolfWooir* announced to the public a discovery respecting)" 11 the expansibility of steam, which promises to be of very essential utility. Mr. Watt had former- ly ascertained, that steam which acts with the expansive force of 4 pounds per square inch, against a safety valve exposed to the weight of the atmosphere, after expanding itself to four times the volume it thus occupies, is still equal to the pressure of the atmosphere. But Mr. Woolf has gone much farther, and has proved, that quantities of steam, having the force of 5, 6, 7, 8, 9, 1O, &c. pounds on. every square inch, may be allowed to expand 5, 6, 7, 8, 9, 10, &c. times its volume, and will still be equal to the atmosphere's weight, provided that the cylinder in which the expansion takes place,. has the same temperature as the steam before it began to ex- pand. It is evident, however, that an increase of temperature is necessary both to produce and to maintain this augmentation of the steam's ex- pansive force above the pressure of the atmo- sphere. At the temperature of 212 of Fahren- heit, the force of steam is equal only to the pres- sure of the atmosphere, and, in order to give it an additional elastic force of 5 pounds per square inch, the temperature must be increased to about 227y , as is evident from the following table. 408 HYDRAULICS. wooifs Table of the Pressures, Temperatures, and Ex- ment. VC pansibility of Steam, equal to the Force of -the Atmosphere. Elastic Force of Steam predominating overthe Pressure of the Atmo- sphere, and acting upon a Safety Valve. Degrees of 1 emp'.- rature requisite for bringing the Steam 'o the different Ex- pansive Forces in the preceding Column. N of times its Voi 1 - that Steam of the preceding G orce and Temperature willexpand,and --till con- tinue equal to the press- ure of the Atmosphere. Pounds ppr square inch Degrees of Heat. Expansibility. 5 227-^ 5 6 230^ 6 7 232^ 7 8 235 8 9 t ''0ii. f IV '-.Ti * 9 10 239J 1O 15 15 20 25Q^ 20 25 ".-. :267 25 30 273 30 , ,( ...,. 35 278 35 40 282 4O In this manner, by small additions of tem- perature, an expansive power may be given to steam, which will enable it to expand 50, 1OO, 2OO, 300, &c. times its volume, and still have the same force as the atmosphere. Upon this principle Mr. Woolf has taken out a patent for various improvements on the steam engine, a short account of which we shall sub- join in the words of the specification. ' ^If the engine be constructed originally with the intention of adopting the preceding improve- ment, it ought to have two steam vessels of dif- ferent dimensions, according to the expansive force to be communicated to the beam j for the HYDRAULICS. 409 smaller steam cylinder must be a measure forwooir the larger. For example, if steam of 40 pounds j the square inch is fixed on, then the smaller steam vessel should be at least part the con- tents of the larger one. Each steam vessel should be furnished with a piston, and the smaller cylinder should have a communication both at its top and bottom, with the boiler which sup- plies the steam, which communications, by means of cocks or valves are to be alternately opened and shut during the working of the engine. The top of the small cylinder should have a commu- nication with the bottom of the larger cylinder, and the bottom of the smaller one with the top of the larger, with proper means to open and shut these alternately by cocks, valves, or any other contrivance. And both the top and bot- tom of the larger cylinder should, while the en- gine is at work, communicate alternately with a condensing vessel, into which a jet of water is admitted to hasten the condensation. Things being thus arranged when the engine is at work, steam of high temperature is admitted from the boiler to act by its elastic force on one side of the smaller piston, while the steam which had last moved it has a communication with the larger cylinder, where it follows the larger piston, now moving towards that end of its cylinder which is open to the condensing vessel. Let both pistons end their stroke at one time, and let us now suppose them both at the top of their respective cylinders ready to descend ; then the steam of 40 pounds the square inch, entering above the smaller piston, will carry it down- wards, while the steam below it, instead of being allowed to escape into the atmosphere, or applied to any other purposes, will pass into the larger 410 HYDRAULICS. cylinder above its piston, which will take its downward stroke at the same time that the pis- ton of the smaller cylinder is doing the same thing ; and, while this goes on, the steam which last filled the larger cylinder, in the upward stroke of the engine, will be passing into the condenser, to be condensed in the downward stroke. When the pistons in the smaller and larger cylinder have thus been made to descend to the bottom of their cylinders, then the steam from the boiler is to be shut off from the top, and admitted to the bottom of the smaller cy- linder, and the communication between the bot- tom of the smaller and the top of the larger cylinder is also to be cut off, and the communi- cation to be opened between the top of the smaller and the bottom of the larger cylinder ; the steam which, in the downward stroke of the engine, filled the larger cylinder, being now open to the condenser, and the communication between the bottom of the larger cylinder and the condenser cut off; and so on alternately, ad- mitting the steam to the different sides of the smaller piston, while the steam last admitted in- to the smaller cylinder passes alternately to the different sides of the larger piston in the larger cylinder, the top and bottom of which are made to communicate alternately with the condenser. * In an engine where these improvements are adopted, that waste of steam which arises in other engines, from steam passing the piston, is totally prevented, for the steam which passes the piston in the smaller cylinder is received into the larger.' Mr. Woolf has also shewn how the preced- ing arrangement may be altered, and has point- ed out various other modifications of his inven- HYDRAULICS. 411 4ion, and the method of applying his improve- ments to steam engines which are already con- structed. On the Power of Steam Engines, and the Method of Computing it. From the account which has been given the steam engine, and the mode of its opera- tion, it must be evident that its power depends upon the breadth and height of the cylinder, or, in other words, on the area of the piston and the length of its stroke. If we suppose that no force is lost in overcoming the inertia of the great beam, and that the lever by which the power acts is equal to the lever of resistance ; then, if steam of a certain elastic force is admit- P LATE ted above the piston q, so as to press it down- % * wards with a force of a little more than 100 pounds, it will be able to raise a weight of 100 pounds hanging at the end of the great beam. When the piston has descended to the bottom of the cylinder, through the space of 4 feet, the weight will have risen through the same space; and 1OO pounds raised through the height of 4 feet, during one descent of the piston, will express the mechanical power of the engine. But if the area of the piston ^, and the length of the cylinder are doubled, while the expansive force of the steam, and the time of the piston's descent remain the same, the mechanical energy of the engine will be quadruple, and will be represented by 200 pounds raised through the space of 8 feet during the time of the piston's descent. The power of steam engines, there- fore, is, cceteris paribus, in the compound ratio of 412 HYDRAULICS. the area of the piston, and the length of the stroke. These observations being premised, it will be easy to compute the power of steam en- gines of any size. Method of Thus, let it be required to determine the p 0wer O f steam engines, whose cylinder is 24 inches diameter, and which make 22 double strokes in a minute, each stroke being 5 feet long, and the force of the steam being equal to a pressure of 12 pounds avoirdupois upon every square inch. 1 The diameter of the pis- ton being multiplied by its circumference, and divided by 4, will give its area in square inches ; 24X3.1410X24 ,, u f thus, - -. == 452.4, the number of 4 square inches exposed to the pressure of the steam. Now if we multiply this area by 12 pounds, the pressure upon every square inch, we will have 452.4X12~5428.8 pounds, the whole pressure upon the piston, or the weight which the engine is capable of raising. But since the engine performs 22 double strokes, 5 feet long, in a minute, the piston must move through 22x5x2220 feet in the same time ; and therefore the power of the engine will be re- presented by 5428.8 pounds avoirdupois, raised through 220 feet in a minute, or by 10.4 hogs- heads of water, ale measure, raised through the same height in the same time. Now this is equivalent to 5428.8X220=1194336 pounds, The working pressure is generally reckoned at 1O pounds on every circular inch, and Smeaton makes it only 7 pounds on every circular inch. The effective pressure which we have adopted is between these extremes, being equivalent to 0.42 pounds on every circular inch. HYDRAULICS. 413 or 1O.4X220 2288 hogsheads raised through the height of 1 foot in a minute. This is the most unequivocal expression of the mechanical power of any machine whatever, that can possi- bly be obtained. But as steam engines were sul> stituted in the room of horses, it has been cus- tomary to calculate their mechanical energy in horse poiuers, or to find the number of hordes which could perform the same work. This in- deed is a very vague expression of power, on ac- count of the different degrees of strength which different horses possess. But still, when we are told that a steam engine is equal to 16 horses, we have a more distinct conception of its power, than when we are informed that it is capable of raising a number of pounds through a certain space in a certain time. Messrs. Watt and Boulton suppose a horse Horse capable of raising 32,OOO pounds avoirdupois P wef *- 1 foot high in a minute, while Dr. Desaguliers makes it 27,5OO pounds, and Mr. Smeaton only 22,9 16. If we divide, therefore, the number of pounds which any engine can raise 1 foot high in a minute, by these three numbers, each quo- tient will represent the number of horses to which the engine is equivalent. Thus, in the present example * \ I * * ? 6 - ~ 37j horses accord- ing to Watt and Boulton ; ^ 9 7 **-; 6 43f horses, according to Desaguliers ; and * 11*11^ n: 52f horses, according to Smeaton. In this calculation, it is supposed that the engine works only eight hours a-day ; so that if it wrought during the- whole 24 hours, it would be equiva- lent to iKrfce the number of horses found by the preceding rule. We cannot help observing, and it is with sincere pleasure that we pay that tribute of respect to the honour and integrity of 414 HYDRAULICS. Messrs. Watt and Boulton which has everywhere been paid to their talents and genius, that in estimating the power of a horse, they have as- signed a value the most unfavourable to their own interests. While Mr. Smeaton and Dr. Desaguliers would have made the engine in the preceding example equivalent to 52 or 43 horses, the patentees themselves state that it will per- form the work only of 37. How unlike is this conduct to some of our modern inventors, who ascribe powers to their machines which cannot possibly belong. to them, and employ the mean- est arts for ensnaring the publicist -^ Perform. Before concluding this article, we shall state watt'f the performance of some of these engines, as steam en- determined by experiment. An engine whose cylinder is 31 inches in diameter, and which makes 17 double strokes per minute, is equi- valent to 40 horses, working day and night, and burns 11,000 pounds of Staffordshire coal per day. When the cylinder is 19 inches, and the engine makes 25 strokes of 4 feet each per mi- nute, its power is equal to that of 12 horses working constantly, and burns '3700 pounds of coals per day. And a cylinder of 24 inches which makes 22 strokes of 5 feet each, performs the work of 2O horses, working constantly, and burns 5500 pounds of coals. Mr. Boulton has estimated their performance in a different manner. He states, that 1 bushel of Newcastle coals, con- taining 84 pounds, will raise 30 million pounds 1 foot high ; that it will grind and dress 1 1 bushels of wheat ; that it will slit and draw into nails 5 cwt. of iron ; that it will drive 100O cot- ton spindles, with all the preparation machinery, with the proper velocity ; and that these effects are equivalent to the work of 10 horses. 415 ' . .>;, HYDRAULICS /srn tifi to 7tbnci;-> t^steing aih Jjsih. 't^lrio nl hiyfe--*fcw--:-jiifo .AVA fao^v. cdi ojjai irjvrtb sM, DESCRIPTION OF A WATER-BLOWING MACHINE. <*<<> :. ...totf J^9 m consequence of the agitation and percussion of its parts. But M. Venturi, 1 to whom we are indebted for the first philosophical account of this machine, has shewn that this opinion is erroneous, and that the wind is supplied from the atmosphere ; for when the lateral openings m, n 9 o, p, were shut, no wind was generated. Hence the principal object in the construction of these machines is to combine as much air as possible with the descending current. With this view the water is often made to pass through a kind of cullendar placed in the open air, and perforated with a great number of small trian- * "fife t't* '-. ^r> *\r\ < -v'<~' 1 Experimental Enquiry concerning the lateral commu- nication of motion in Fluids. Prop. 8. HYDRAULICS. 41*7 gular holes. Through these apertures the water descends in many small streams, and by exposing a greater surface to the atmosphere, it carries along with it an immense quantity of air, and is conveyed to the pedestal P by a tube C H, open and enlarged at (7, so as to be considerably wider than the end of the pipe which holds the cul- lendar. It has been generally supposed that the water- fall should be very high ;"* but Dr. Lewis has shewn, by a variety of experiments, that a fall of 4 or 5 feet is sufficient, and that when the height is greater than this, two or more blow- ing machines may be erected, by conducting the water from which the air is extricated into ano- ther reservoir, from which it again descends and generates air as formerly. That the air, which is necessarily loaded with moisture, may arrive at the furnace in as dry a state as possible, the condensing vessel D E should be made as high as circumstances will permit ; and in order to determine the strength of the blast, it should be furnished with a gage a b filled with water. Franciscus Tertius de Lanis observes, 3 that he has seen a greater wind generated by a blow- ing machine of this kind, than could be produced by bellows 10 or 12 feet long. The rain wind is produced in the same way as Cause of the blast of air in water-blowing machines. When the drops of rain impinge upon the surface of the sea, the air which they drag along with them often produces a heavy squall, which is sufficient- ly strong to carry away the mast of a ship. The 1 Wolfius makes the length of the tube C H 5 or 6 feet Opera Mathematics, torn, i, p. 830. } In Magisterio Naturae et Artis, lib. v, cap. .1. Vol. IL D d 418 HYDRAULICS. same phenomena happens at land, when the clouds empty themselves in alternate showers. In this case, the wind proceeds from that quar- ter of the horizon where the shower is falling. The common method of accounting for the ori- gin of winds by local rarefactions of the air, ap- pears to me pregnant with insuperable difficulties; and I am apt to think, that these agitarions in our atmosphere, ought rather to be referred to the principle which we have now been consi- dering. 4 4 Those who wish for more information upon the subject of water-blowing machines, may consult Lewis's Commerce of Arts ; the Journal des Mines, N pi ; or Nicholson's Journal, vol. xii, p. 48. I -*.^, LAMPREY, RENDELL & LAM! t PLUMBERS, tc Engine U, BACHELQR'S WALK, Dublin IMPROVED HYDRAULIC RAM. T>ESPECTFULLY beg to call the attention of GENTLEMEN, ARCHITECTS, and I It Improved Self-acting WATER RAM, which (without manual labour) will raise wa- 300 feet, or thirty times the height of the fall by which it is worked. THE HYDRAULIC RAM cannot be applied to stagnant water; a fall is absolutely ne spring or brook have not a DAM HEAD already, one may be constructed, in most situatioi Parties who have small running streams on their estates, and require water to be raise. Houses or other Buildings, may be furnished with estimates of the expense of obtaining it, a a certainty of success, by giving Messrs. LAMPREY, RENDELL, & CO. the following in 1st The number of feet of fall they can obtain at the Springer Brook to work the Ram. 2nd The perpendicular height from the lower part of the fall to where the water is requir 3rd The distance, horizontally, from the spring or brook to the house or premises where t 4th. If the spring or stream should be small, it is very desirable to ascertain how many minute. LAMPREY, RENDELL, & CO. fit up all apparatus connected with heating CO) MANSIONS, &c., either by STEAM or HOT WATER. N.B The following Noblemen and Gentlemen have had these Engines erected: Lord Rossmore. Lord Gough. Sir George Hodson, Bart. Sir G. Palmer, Bart. , Judge Moore. George Roe, Esq., J. Arthur Murray, Esq. Charles Tottenham, I .lll'ln. I IE) .DERS to their o the height of ary. But if the : a small outlay, the top of their Iso be put upon ation : > be delivered, ater is wanted. 3ns it flows per RVATORIES, HYDRAULICS. WHITEHURST'S MACHINE FOR I BY ITS MOMENTUM; AND MONT- IAULIC RAM. idea of raising water by the mo- The idea of ater itself was first suggested bv raising by I ; 4-1, r>U'i u- P>n 7 Jts momen- m the rnilosopmcal I ransac- tum su g - The same principle, in an im- * ested b ? ,1.1, j TI Mr.White- 5 lately been revived in France, hurst. considerable attention both on d in this country. Whatever j is due to the inventor of the roperly belongs to our country- urst, and Montgolfier can lay more than the merit of an im- st's machine, which is represent- Description 'Plate XIV, was actually erect- wh ". Cheshire, and completely an- hursfs ma- ration of its inventor. AM isp 1 ^', /oir, whose surface is on a level xiv, >m of the reservoir B N. s If inches diameter, and near- ards long, and the branch pipe .j, size, that the cock F is about Dd 2 419 HYDRAULICS. DESCRIPTION OF WHITF.HURST'S MACHINE FOR RAISING WATER BY ITS MOMENTUM; AND MONT- GOLFIER'S HYDRAULIC RAM. JL HE ingenious idea of raising water by the mo- The idea of mentum of the water itself was first suggested by raisin e b ? oo / its tnomen- Mr. Whitehurst in the Philosophical Transac- tum sug- tions for 1775. The same principle, in an im- J proved form, has lately been revived in France, hurst, and has excited considerable attention both on the continent and in this country. Whatever credit, therefore, is due to the inventor of the hydraulic ram, properly belongs to our country- man Mr. Whitehurst, and Montgolfier can lay claim to nothing more than the merit of an im- prover. Mr. Whitehurst 's machine, which is represent- Description ed in figure 2 d of Plate XIV, was actually erect- white- ed at Oulton in Cheshire, and completely an- hurst's ma- swered the expectation of its inventor. AM isp^' the original reservoir, whose surface is on a level xiv, with B, the bottom of the reservoir B N. The* main pipe AE, is 1^ inches diameter, and near- ly two hundred yards long, and the branch pipe F is of such a size, that the cock F is about Dd 2 42O HYDRAULICS. 1 6 feet below the surface M of the reservoir, D is a valve box, with its valve a, and C is an air vessel, into which are inserted the extremities m, n, of the main pipe, bent downwards to pre- vent the air from being driven out when the wa- ter is forced into it. Now, since the difference of level between the cock F, and the top of the reservoir A M is Id feet, upon opening the cock F the water will rush out with a velocity of nearly 3O feet per second. A column of water, therefore, two hundred yards long, is thus put in motion, and, though the aperture of the cock F be small, it must have a very considerable momentum. Let the cock F be now suddenly stopped, the water must evidently rush through the valve a into the air vessel C, and condense the included air. This condensation must take place every time the cock is opened and shut, and the included air being highly compressed, will press upon the water in the air vessel, and raise it into the reservoir B N. Description From this brief description of Whitehurst's goifier N engine, the reader will easily perceive its resem- hydrauiic blance to that of Montgolfier, a section of PLATE which is represented by figure 3 d . R is the re- xiv, servoir, R S the height of the fall, and S T the Fl S- 3- A PP- horizontal tube which conducts the water to the engine A B H T C. E and D are two valves, and FG a pipe reaching within a very little of the bottom C B. Now let water descend from the reservoir, it will rush out at the aperture m n till its velocity becomes so great as to force up the valve E. The water being thus suddenly checked, and unable to find a passage at mn, will rush forwards towards H 9 and raise the valve D. A portion of water being admitted into the vessel A E C, the impulse of the column of HYDRAULICS. 421 fluid is spent, the valves D and E fall, and the water rushes out at m n as before, when its motion is again stopped, and the same operation repeated which has now been described. Every time therefore that the valve E closes, a portion of water will force its way into the vessel ABC* . and condense the air which it contains ; for the included air has no communication with the at- mosphere after the water is higher than the bottom of the pipe FG. This condensed air will consequently exert great force upon the sur- face op of the water, and raise it in the tube FG to a height proportioned to the elasticity of the imprisoned air. The external appearance of the engine, copied from one in the possession of Professor Leslie, is exhibited in Figure 4, where Fig, 4, A B Cis the air vessel, Fthe valve box, G the extremity of the valve, and M, N, screws for fixing the horizontal tube to the machine. A piece of brass A, with a small aperture, is screwed on the top when the engine is employed to form a jet of water. From this description, the read- er will perceive, that the oniy difference between the engines of Montgolfier and Whitehurst is, that the one requires a person to turn the cock, while the other has the advantage of acting spon- taneously. Montgolfier indeed observes, that the honour of this invention was not due to Eng- land, but that he was the sole inventor, and did not receive a hint from any person whatever. * We leave it to the reader to determine what degree of credit these assertions are entitled to. 1 * Cette invention n'est point originairc d'Angleterre, die appartient toute entierc a la France ; je declare quc j'co suis le seul invcntcur, et que 1'idte nc m'en a etc fournic par personne.' Journal det M'met^ vol. xiii, No. 73. Dd3 422 HYDRAULICS. It would appear from some experiments of Montgolfier, that the effect of the water-ram is equal to between j- and \ of the power ex- pended, which renders it superior to most hy- draulic machines.* * Those who wish for more information on this sub- ject, may consult the Journal des Mines, vol. xiii, No. 73, where Montgolfier has given a very unphilosophical ac- count of the ram. Sonini's Journal, Feb. 18O6, p. 334, and Nicholson's Jpurnal, No. 56, vol. xiv, p. 98. 423 OPTICS. ON ACHROMATIC TELESCOPES. On Achromatic Object Glasses. JN or WITHSTANDING the claims of foreigners to the invention of the achromatic telescope, we have the most unquestionable evidence that this instrument was first invented and constructed about the year 1758, by our countryman Mr. John Dollond. As telescopes of this description are not affected with the prismatic colours, the late Dr. Bevis proposed to distinguish them by the name of Achromatic, an appellation which they have hitherto retained, though some have erroneously stated that it was first given them by the astronomer Lalande. During the 17 th century, when every branch of science was cultivated with unwearied assi- duity, the attention of philosophers was particu- larly directed to the improvement of the refract- ing telescope. But as the different refrangibility of the rays of light was then unknown, men of science employed themselves chiefly in trying to remove the spherical aberration, or the error which arises from the spherical figure of the ob- 424 OPTICS. imperfec- ject glass. For this purpose they ground their tions of te- o bject lenses of a parabolic or hyperbolic figure. Icscopcs. J i_ i r v t * j r i or or a spherical form, with the radius or the sur- face next the object six times greater than that of the Surface next the eye, in which case Huy- gens had shewn that the aberration is less than when the radii of curvature have any other pro- portion. After all these trials, however, the re- fracting telescope still retained its former imper- fections, and the opticians of those days, com- pletely despairing of bringing it to perfection, turned the whole of their attention to the con- struction of the reflecting telescope. It was reserved for Sir Isaac Newton to dis- cover the cause of these imperfections, and for SJtSfTi Dollond to point out their cure. From Newton's of Dollond. _, r *1 . . . . . . . Theory or Colours, it plainly appeared that the imperfections of the dioptric telescope arose from, the different refrangibility of the rays of light, and that, compared with this, the spherical aberration was extremely trifling. But though Newton, by thus pointing out the cause of the indistinctness of refracting telescopes, contri- buted indirectly to their improvement, he may certainly be said to have checked the progress of this branch of science, when he stated, 4 ' that all refracting substances diverged the prismatic colours in a constant proportion to their mean refractions, that refraction could not be pro- duced without colour, and, consequently, that no improvement could be expected in the re- fracting telescope.' In this conclusion philo- sophers had acquiesced for above half a century, till Mr. Dollond having examined the premises Newton's Optics, p. 112. OPTICS. 425 from which it was drawn, obtained a result very different from that of Sir Isaac Newton. He found that substances which had the same refrac- tive power had different dispersive powers, or, in his own words, 1 ' that there is a difference in ' the dispersion of the colours of light, when e the mean rays are equally refracted by different * mediums ;' and thence concluded that the ob- ject glasses of refracting telescopes were capable of being made, without being affected by the different refrangibility of the rays of light. That our readers may understand this illus- trious discovery, and the method of its applica- tion to the construction of achromatic object glasses, let AE C be a prism, a beam of white P . LATE IX light proceeding in the direction ON, but re- Flg * 5 ' Ap ?* fracted from its rectilineal course by the inter- position of the prism, and forming the prismatic spectrum R MV. The line n M being the direc- tion of the mean refracted light, the angle Nn M is called the angle of deviation, and It n P r the a7eg/e of dissipation, or dispersion. In the same medium, the angle of dispersion is always pro- portional to the angle of deviation, or to the mean angle of refraction, and Newton imagined that this was universally the case in different me- dia, i. e. that the angle Nn M is always propor- tional to the angle R n ^, whether the prism be of crown, or flint, or any other kind of glass. Dollond, however, found that these angles were not proportional to each other in different media^ but that in some the angle of deviation is larger when the angle of dispersion is smaller, and that in others the angle of deviation is smaller when 1 See the Philosophical Transactions, ol. 50, p. 743. 426 OPTICS. the angle of dispersion is larger. Thus, if the prism A B C be made of crown glass, the angle of deviation will be Nn M, and that of dispersion RnV, but if a flint-glass prism with a less re- fracting angle be substituted in its room, the angle of deviation may be Nn M, while that of dispersion becomes r n ?;. The application of these principles to the im- ach- provement of the refracting telescope will be telescopes. eas *ty comprehended, if we consider that light is refracted and dispersed by lenses in the same PLATE ix manner as ^Y P r i sm s. Thus, in. Fig. 6, let A B Fig. 6. ' be a convex lens, a beam of light incident at 77z, emerging at m, and proceeding in the direc- tion m JV ; then if we suppose a b c to be a prism, whose sides a b y a having the same curvature and the same refractive and dispersive power, then the ray of white light 0, incident at 72, will emerge colourless at p 9 and proceed in the direction p N, parallel to n, because the change which is produced on the incident ray by the first prism or convex lens, is counteracted by an equal and opposite change produced by the second prism or concave lens. But if the second prism or lens has a different refractive and dis- persive power from the first, and if the refract- OPTICS, 427 ing angle of both is the same, the ray p N will be coloured after refraction, because the second prism more than counteracts the effects of the first, and it will be bent to or from the axis of the lenses according as the refracting power of the second prism or lens is greater or less than that of the first. From these observations, the attentive reader Double will easily understand the construction of the a ^ ro " 1 f lc i i_*i_/* j-> object glass, double achromatic object glass, in which A B Fig. 6. is the convex lens of crown glass, and CD the concave one of flint glass. As the re- fractive and dispersive powers of the lens CD is greater than those of the lens A J5, the curva- tures of the lenses, or the refracting angles of the corresponding prisms, being equal, the ray P N will be bent from the axis of the lenses, and it will be coloured by the excess of the dispersive power of the flint above that of the crown glass. In this case, therefore, the combined lenses will not have a positive focus. But, since, in the same medium, the angle of dispersion increases or diminishes with the angle of deviation, we can diminish the refraction and dispersion of the con- cave lens by diminishing its concavity, or the re- fracting angle of the corresponding prism. Now, let the concavity of the lens CD be diminished till its dispersion be equal to the dispersion of the lens A B, its refraction or power of bending the incident rays will also be diminished ; then since the dispersion of the concave lens is equal to the dispersion of the convex one, and its cur- vature less, the ray p N will emerge perfectly colourless, and it will be bent towards the axis of the lenses, as the convergency of the incident ray occasioned by the convex lens is not wholly counteracted by the divergency produced by the 428 OPTICS. concave one. In the same mann^" every other ray falling upon the surface of si B will be re- fracted colourless into a positive focus, and an image \vili be formed perfectly achromatic. Triple ach- jr rom v; h a t has been said concerning the double romaticob- ... i -n i ject gias. achromatic object glass, it will be easy to com- prehend how a colourless image is formed by a combination of three lenses, which is now uni- versally adopted for the purpose of diminishing Plat* ix, t he spherical aberration. In Fig. 7, let A B 9 CD, E F, be the three lenses which compose the triple object glass, 3 AE^ and E F, being convex, and of crown glass, and CD concave, and made of flint glass j and let a b, c d, ef, be corresponding prisms, which, if substituted in- stead of the lenses, would refract and disperse, in a similar manner, any ray of light which falls upon the points q, m, r, t, g, p, where the sides of the prisms are supposed to touch the surfaces of the lenses. Suppose, also, w r hich is generally the case, that the two convex lenses have equal focal lengths, and that the focal distance of either lens is greater, or their curvature less, than that of the concave one, whose dispersion exceeds that of the lens A B ; then a ray, O, of white light incident at q 9 will, after refraction by the lens AB, be separated into its compo- nent parts, and proceed in the direction m r R 9 n s V ; m r R being the extreme red ray, and n s ^the extreme violet. But as these rays are intercepted by the lens CD, at the points r, s, they will undergo another refraction in a con- trary direction, and will proceed according to 3 The lenses are placed at a distance from each other in the figure, that the progress of the incident ray may be more easily perceived. OPTICS. 429 the dotted lines t r, o v. These rays will diverge after refraction, and be bent from the axis of* the lenses, since the refraction, as well as the re- fracting angle of the prism c d 9 or lens c D 9 ex* ceeds the refraction, and refracting angle of the prisni a b, or lens A B ; for though the violet ray q n s is bent from the red ray q m r, by the refraction of the lens A B, it is again bent to- wards it by the superior refraction of the concave lens, and they will therefore converge to one another in the direction tr, ov. In this case, the excess of dispersive power in the concave lens tends only to delay the mutual convergency of the red and violet rays, or to remove the point where they would meet farther from the lens e D. Now it is evident that two rays of different refrangi- bility falling upon a prism or lens with different angles of incidence, may emerge with the same angle of refraction, or may be united at their emersion from the prism or lens; for in this case their difference of refrangibility counteracts the difference between their angles of incidence. The red and violet rays or, I v, therefore, which fall upon the lens E F, with different angles of incidence, will, after refraction by the third lens, proceed perfectly colourless in the direction pN. In the same manner, all the rays which proceed from any object, emerging colourless from the triple object-glass, will unite in one point, and form an image completely achromatic. Having thus discovered that light could be re- fracted without colour, the next object of philo- sophers was to ascertain the curvature which must be given to lenses, in order to produce this effect, and likewise to correct the spherical aber- ration. This subject has been treated with the greatest ability by several foreign mathematicians, 430 OPTICS. but particularly by Euler, 1 D'Alembert,* Clai- raut, 3 Boscovich, 4 and Klugel. 5 The writings of these philosophers furnish us with the most complete and accurate information upon this point ; and art has in this case received from science all the assistance which she can possibly bestow. It shall be our object at present to re- duce the results of their investigations either into tables or into such a form as may be easily com- prehended by the practical optician, and thus to furnish the artist with a popular view of this in- teresting subject. For this purpose, the cele- brated Euler has given, in his Dioptrics, 6 two formulae, from which we have calculated the two following tables, containing the radii of curvature for the lenses of a triple object-glass. The first column contains the focal distance of the lenses when combined ; and the six following columns contain the radii of their curvature in inches and decimals, beginning with the surface next the object. 1 Comment. Nov. Acad. Petropol. Tom. 18, p. 407- a Mem. de 1'Acad. Paris, 1764, 8 VO , p. I3g ; 1765, 8 VO , p. 81, and 1767, 4% p. 43. * Mem. de 1'Acad. Paris 1756, 8 VO , p. 6l2 ; 1757, 8 V % p. 853, and 1762. 4 R. J. Boscovichii Opera pertinentia ad Opticam et Astronomiam, Bassani. 1785, Tom. 1. Opusc. 2, p. 169. 5 Comment. Reg. Soc. Getting. 1795 to 1798, Tom. 13, p. 28. 6 The mean refraction of the crown gkss is supposed to be 1.53, and that of the flint glass 1.58, and their disper- ve powers as 2 to 3. OPTICS, 431 TABLE I. Talk of the R ^Jii of Curvature of the Tenses of a Triple Ach- romatic Object Glass, according to Baler's first Formula. .rocal ength. Convex lens of Crown Glass. Concave kn> of Flint Glass. Convex lens of Crown Glass ~> .in A >erture. Inch. Inches. inches. inches. Inch-s. Inches. Inches. Inches. 6 30O 22 OO 3.10 2.91 3.13 2.85 0.71 9 4.50 33.OO 4.65 4.36 4.70 4.28 1.O7 12 6.00 44.00 6. 2O 5.82 6.26 5.7O 1.43 18 24 900 12.00 66.0O 88.00 9.30 12.40 8.72 11.64 9.4O 12.52 8.56 f 1 .40 2.14 2.86 3O 15.01 109 99 15.5O 14.53 15.65 14.27 3.56 36 18.O1 131.99 18.60 17.44 18.80 17.12 4.28 42 21.01 153.9921.70 20.37 21.91 19.97 4.99 48 24.02 175.99 24.80 23.28 25.04 22.80 5.72 54 60 27.O2197.99 30.02219.99 27.90 31.OO 26.16 29.06 28.18 31.31 25.69 '28.54 6.42 7.13 Tables for triple ob- ject glasses TABLE II. Talk of the Radii of Curvature ofth: Lenses of a Triple Ach* romatic Object Glass, according to Euler's second Formula. "ocal len lh . Convex lens of Crown Glass. Concave lens of Flint Glass. Convex lens of Crown Glass. Semi A- icrture, Inch. Inches. Inches. Inches. Inches. Inches. Inches. Inches. 6 1.70 12.44 12.88 1.77 3.56 15.00 0.42 9 2.55 18.66 19.31 2.66 5.34 22.51 O.64 12 3.40 24.87 25.75 3.55 7.13 3O.01 0.85 18 5.O9 37.31 38.63 5.32 10.69 45.O1 1.27 24 6.80 49.74 51.50 7.10 14.25 6O.O2 1.70 3O 8.49 62.19 64.38 8.86 17.81 75.02 2.12 36 10.19 74.63 77.26 1O.63 21.37 90.02 2.54 42 11.89 87.07 90.14 12.40 24.93 1O5.03 2.96 48 13.60 99.48 103.O2 14.17 28.49 120.03 3.39 54 15.27 I 11. 93*1 15.87 15.96 32.07 135.04 3.82 60 16.97 124.37jl28.75 17.73 35.63 1 50.04 4.24 OPTICS* The only person in our country who has writ- ten upon the theory of achromatic object glasses, is the late learned Dr. Robison of Edinburgh, who, following the steps of Clairaut and Bos- covich, has given an interesting dissertation upon this subject. 7 From the formulae contained in that dissertation, the following table is computed. TABLE III* Table of the RaJii of Curvature of the Lenses of a Triple Object Glass. Focal ength. Convex lens of Crown glass. Convex lens of Flint glass* Convex lens of Crown glass. Inches- Inches. Inches Inches. Inches. Inches. Inches. 6 4.54 3.03 3.03 6.36 6.36 0.64 9 6.83 4.56 4.56 9.54 9.54 0.92 12 9.25 6.17 6.17 12.75 12.75 1.28 18 13.67 9.12 9.12 19.08 19.08 1.92 24 18.33 12.25 12.25 25.50 25.50 2.56 30 22.71 15.16 T5.16 31.79 31.79 3.20 36 27.33 18.25 18.25 38.17 38.17 3.84 42 31.87 21.28 21.28 44.53 4453 4.48 48 36.42 24.33 24.33 50.92 .50.92 5.12 54 40.96 27.36 27.36 57.28 57.28 5.76 60 45.42 3O.33 3O.33 63.58 63.58 6. 4 The reader will observe, that only three pair of grinding tools are necessary for constructing a telescope according to the preceding table ; but the work may be performed by only two grind- ing tools, if we employ the radii of curvature, 7 Article Telescope, Encyclopaedia Britannica, vol. xviii, p. 338. 8 A telescope 3O inches in focal length, constructed ac- cording to this table, bore an aperture of 3f inches. OPTICS*' 433 which are contained in the following table, com- puted from the formulae of Boscovich. TABLE IV. Focal length. Radii of the four surfaces of the two lenses of Crown glass. Radius of the two surfaces of the concave lens of Flint glass. Inches. Inches. Inches. 6 3.84 3.17 9 5.76 4.75 12 7.68 6.34 18 11.52 9.5O 24 15.36 12.68 30 19.2O 15.84 36 23.04 19.00 42 26.88 23.17 48 30.72 25.36 54 34.66 28.51 60 | 38.40 31.68 TABLE F. The Radii of Curvature employed by the London Opticians are pretty nearly represented in the following Table , which is col* culatedfrom Dr. Robison's Measurements. Focal length. Convex lens of Crown glass. Radius of botn tne sur- faces of the concave lens of Flint elass. Convex lens of Crown glass. Inches. Inches. Inches. Inches. Inches. Inches* 6 3.77 4.49 3.47 3.77 4.49 9 5.65 6.74 5.21 5.65 6.74 12 18 24 7.54 11. SO 15.08 8.99 13.48 17.98 6.95 10.42 13.9O 7.54 11. 3O 15.08 8.99 13.48 17.98 30 18.34 22.47 17.37 18.34 22.47 36 22.61 26.96 20.84 22.61 26.96 42 26.38 31.45 24.31 26.38 31.45 48 30.16 35.96 27.80 30.16 35.96 54 33.91 40.45 31.27 33.91 40.45 60 37.68 44.94 34.74 37.68 44.94 Ee 434 s OPTICS. Radii of Two of Dollond's best achromatic telescopes inDoliond's being examined, were found to have their lenses telescopes, of the following curvatures, reckoning from the surface next the object. Crown glass lens, 28 inches and 40. Concave lens 2O.9 inches, and 28. Crown glass lens 28.4, and 28.4. The focal length of the compound object-glass was 46 inches. In the other telescope, whose focal length was 46.3 inches, the curvature of the 1 st lens was 28 and 35.5 inches; the, 2 d lens 21.1 and 25.75; and the 3 d 28 and 28. Both these telescopes mag- nified from 10O to 20O times, according to the powers applied. The due de Chaulnes having in his possession one of* Dollond's best telescopes, whose focal length was 3 feet 5 inches 4.25 lines, made a va- riety of accurate experiments in order to deter- mine the curvature, thickness, and distance, of its lenses, and found them to be of the following dimensions. 5 Radius of the l ft surface, or the surface next the object, 25 inches 11.5 lines. Radius of the 2 d surface 32 inches 8 lines. Ra- dius of the 3 d surface 17 inches 10 lines. Ra- dius of the 4 th surface 24y inches. Radius of the 5 th 241 inches ; and the radius of the 6 th 26 inches and 10.6 lines. Thickness of the first lens at its axis 2. 1 1 lines ; thickness of the 2 d 1.59 lines; thickness of the 3 d 2.18; and the thickness of the whole lens 5.91 lines. 6 A very excellent telescope, with a double achromatic object glass, was constructed by M. Antheaulmein 1763, from the formulae of Glair- 7 These experiments are detailed at great length in the Mem. de 1'Acad. Paris, 1767, 4 l<> , p. 423. 6 For the dimensions of the eye piece of this telescope, see the article on Achromatic Eye pieces, p. 458. OPTICS. 435 aut. The lens of flint glass was placed next the object, and was a meniscus with its convex side outwards. The radius of its concavity was 17-^ inches, and the radius of its convex side was 7 feet 64- inches. The interior surface of the lens of crown glass had a radius of 1 8 inches, while its exterior surface, or that next the eye, was 7 feet 6 inches. These lenses were separated by a piece of card, and formed a compound object glass, with a focal length of 7 feet, and an aperture of 3 inches and 4> lines. Its eye-piece consisted, of two lenses. That next the object was a double con- vex lens with 1 8 lines of focal length, and 9 lines of aperture. Its first surface, or that next the 'ob- ject glass, had a radius of 1 1-^- lines; and its se- cond surface a radius of 7 inches 2 lines. - The second eye-glass, which was a meniscus, had 5 lines of focus and 2 lines of aperture. The ra-. dius of its convex surface was 2j lines, and that' of its concave surface, which was next the eye, was 8 lines. The distance between the two eye- glasses was 9 lines. " 5 " "** r - i .-id im/ft bnft (ii'n 436 OPTICS. Tables for Double Achromatic Object Glasses. Tables for The following table, calculated from the for- double ob- mulae of Boscovich, contains the radii of cur- es vature for the lenses of a double achromatic ob- iect glass. TABLE vi. Focal length. Convex Lens of Crown Glass. Concave Lens of Flint Glass. Inches. Inches. Inches. Inche*. Inches, 6 1.94 1.91 1.91 9.49 9 2.91 2.86 2.86 14.24 12 3.88 3.82 3.82 18.99 18 5.82 5.73 5.73 28.48 24 7.76 7.63 7.63 36.99 SO 9.70 9.54 9.54 47.47 36 11.64 11.45 11.45 56.97 42 13.58 13.36 13.36 66.46 48 15.51 15.27 15.27 73.98 54 17.45 17.17 17.17 85.47 60 19.39 19.08 19.08 94.95 70 22.62 22.26 22.26 110.77 80 25.86 25.44 25.44 126.60 90 29.09 28.62 28.62 142.42 1OO 32.32 31.80 31. 8O 158.25 In the following table, calculated from Dr, Robison's measurements, the reader will find the radii of curvature which are employed by the London artists in the construction of the double achromatic object glass. OPTICS. 437 TABLE VII.. Focal length. Convex Lens of Crown Glass, Concave Lens of Flint Glass. Inches. Inches. Inches. Inches. Inches. 6 1.76 2.12 2.07 6.88 9 2.64 3.17 3.1O 10.33 12 3.53 4.23 4.13 13.77 18 5.29 6.35 6.20 20.65 24 7.05 8.46 8.26 27.54 30 8.81 10.58 1O.33 34.42 36 10.58 12.69 12.39 41.3O 42 12.34 14.81 14.46 48.18 48 14.11 16.92 16.52 55.07 54 15.87 19.04 18.59 61.96 60 17.63 21.16 20.66 68.84 70 20.57 24.68 24.10 80.32 8O 23.50 28.21 27.54 91.79 90 26.44 31.73 30.99 103.27 100 29.38 35.26 34.43 114.74 Although it has been the practice in this coiin- Ta bi e for try to construct only double and triple achroma- quadruple tic object glasses, yet they may be composed ^"3. even of four or five lenses, the convex ones of crown glass, and the concave ones of flint glass being placed in an alternate order. By aug- menting the number of media, indeed, a quan- tity of light must be lost, and the labour of the artist greatly increased ; but M. Jeaurat informs 4 In this and the six preceding tables, the sine of inci- dence is supposed to be to the sine of refraction as 1.526 to 1 in the crown glass, and as 1 .604 to 1 in the flint glass ; and the ratio of the differences of the sines of the extreme rays in the crown and flint glass O.6054. Ee 3 438 OPTICS. us, that he constructed a compound object glass 5 inches and 1O lines in focal length, which bore an aperture of ly inches, while the best achro- matic telescopes of 6 inches focus, constructed in England, had an aperture only of an inch and a quarter. To such, therefore, as wish to con- struct object glasses of this description, the fol- lowing table, containing their radii of curvature, may probably be acceptable. TABLE rni, s Focal length of the compound object glass. Radius of the six interior surfaces. Radius of the two exterior surfaces. Aperture of the object glass Feet. Inches Feet. Inch Dec, Feet. Inch Dec Inch. Dec. O 4 O 3.O8 O 3.58 1.25 6 O 4.50 5.33 1.50 8 5.92 7.08 1.83 10 O 7.33 O 8.83 2.17 1 8.83 10.58 2.25 2 1 5.33 1 9.17 2.58 3 2 1.83 2 7.67 2.92 4 O 2 4.42 3 6.25 3.25 5 3 6.92 4 4.75 3.58 6 4 3.50 5 3.33 3.92 7 5 O.OO 6 1.83 4.17 8 5 8.58 7 0.42 4.50 9 6 5.08 7 11.00 4.83 1O O 7 1.58 8 9.50 5.17 in' the con- Though it is demonstrable that a telescope struction of constructed according to the preceding tables, achromatic | telescopes. s This table supposes that the mean refraction of the crown glass is to that of the flint glass as 1000 to 1045, and their dispersive powers as 200 to 353. OPTICS. 439 and formed of glass, whose refractive and dis- persive power is similar to that which was em- ployed in the formulae upon which these tables are founded, will form an image perfectly dis- tinct and colourless ; yet it is so difficult to pro- cure flint glass of the same refractive and dis- persive power, that it is almost impossible for a private individual to succeed, even after several trials. The London optieians have always at hand a number of lenses of various curvatures, and different powers of refraction and disper- sion, and by selecting such as answer best up- on trial, they are enabled, without much trouble, to construct an object glass in which the sphe- rical and chromatic aberrations are almost wholly corrected. Those, therefore, who are not fur- nished with a sufficient number of lenses, must necessarily meet with, frequent disappointments in their attempts to construct achromatic tele- scopes ; and the only way of preventing these disappointments, and rendering success more cer- tain, is to have a variety of tables, which being founded on different conditions, give different curvatures to the lenses. If the artist should be unsuccessful, either from the nature of the re- fracting media which he employs, or from giving the lenses a greater or lesser curvature than the table requires, he may, with very little trouble, sometimes with altering the radius of a single surface, adapt the lenses to the conditions of some other table, and in all probability obtain a more favourable result. With the view of faci- litating these attempts, we have computed the eight preceding tables, and for the same purpose we shall subjoin the following different forms of achromatic object glasses from Boscovich and Klugel. 44Q OPTICS. In these forms a represents the first surface of the compound lens, or that which is next the object, b the second surface, a' the third, b' the fourth, a" the fifth, and b'' the sixth ; a, b, a", b" representing the radii of curvature for the con- vex lenses of crown glass, and #', b' the curva- ture of the concave lens of flint glass. The focal distance of the first lens, or that whose surfaces are marked a, /;, is represented by x, that of the second by ?/, and that of the third by z, while the focal length of the compound lens is distin- guished by the letter F. I. Forms for =a"=l>"=0.64Q z=x=0.608 rf= =0.529 ^=0.438 II. First Lens Isosceles* ^=^=0.810 z=x=0.6O8 ta>=:&=a"=0.52d ^=0.438 '' If it is required to erect the object as in the Galilean telescope, the middle lens of flint glass must be made convex, and the other lenses con- cave, but with the same radii of curvature, so that the concavity of the compound lens may predominate. On Eye Pieces with three Lenses, which remove the Chromatic Aberration. Eye-pieces The three lenses must be made of the same with three kj n( j o gi ass an ^ may be of any focal length. lenses. , 6 > / 7 & Ine distance between the first and second, or the two next the object, must be equal to the sum of their focal distances, and the distance between the second and third must exceed the sum of their focal distances, by a quantity which is a third proportional to the distance between the first and second, and the focal length of the second lens ; or, in other words, the distance be- tween the second and third lenses must be equal to the sum of their focal distances, added to the quotient arising from the square of the focal dis- tance of the second lens, divided by the sum of the focal distances of the first and second. These, and other circumstances, which should be attend- ed to in the construction of achromatic eye pieces. OPTICS. 447 will be better understood by expressing them algebraically. Thus, let .Fbe the focal length of the object glass, aad a?, ?/, z, the focal distances of these eye glasses, reckoning from that which is nearest the object. Then we shall have 1 , The distance between the first and second jfo""" 1 * < for achro- tenses - _- fl . x-\-y mat i c eye- 2, The distance between the second pieces. v 1 and third - - y4-z4^-~ V I v 3, Distance of the first lens from the focus of the object glass - -^~ x+y 4, Magnifying power of the eye-piece 5, Focal distance of a single lens with the same magnifying power G, Distance of the eye from the third lens - z 7, Length of the whole eye-piece o>f-3?/-{-2z 8, Length of the whole telescope F- 9, Aperture of the lenses' a, a, a\..a a JO, The aperture of the diaphragm, or field bar, or m, should be a little less than - a And should be placed in the fo- cus of the object glass. 1 1 , The field of view is ne&ly ^ ' 1 The apertures of the lenses may be made equal to one another, but should never be greater than half the focal distance of the third lens. xz y 448 optics. Although the aberration of colour will b com- pletely removed by making the lenses of any fo- cal length, and placing them at the distances in- dicated by the preceding formulae, yet it is pre- ferable to make the first and second lenses of the same focal length, and to give the third a less 1 focal distance, and make its distance from the second equal to its own focal length, added to 1 ^ the focal distance of one of the other lenses ; for, in this case, where x and y are equal, the expres- yt- sion -^- , which, when added to y-f-z, expresses the distance between the second and third lenses, becomes ^y. 2 Beside the simplicity of this com- bination, it has another advantage, for the mag- nifying power of the eye-piece is always equal to the magnifying power of the third lens. This is evident from the fifth formula , which becomes zrz when x and y have the same value. So that in this construction, when we wish to give a cer- tain magnifying power to a telescope, we have only to take such a focal length for the third* lens as will produce this magnifying power, and make the focal length of the other two a little greater than that of the third. By increasing the focal lengths of the two first lenses, the image is not injured by any particles of dust which may be lying on their surface, and the spherical aber- ration is also diminished. By augmenting the curvature of the third lens, however, we con- tract the field of view, which ought, if possible, v* y 1 v * Since *=jf in this case, -* is =-="' or \y for 4f. .x-j-jr 2y 2 X 2y=j.. OPTICS. 449 to be avoided. This may be avoided, indeed, as Boscovich has shewn, by making the third lens consist of two convex ones of the same glass, their surfaces being in contact, and their focal lengths equal. From long experience, he found that eye pieces of this construction are superior to all others, and that the error arising from the spherical figure of the glass is greatly diminish- ed, by making all the lenses plano-convex, and turning the plain sides to the eye, excepting the second lens, whose plane surface should be turn- ed to the object. All the lenses may be made of the same focal length, and then the distance between the first and second, and the second and third, will be equal to the sum of their focal dis- tances. In this case the third and fourth lenses, which are joined together, are considered as a single lens, whose focal length is equal to one half the focal length of either of the two. The apertures, too, may be all equal, and the field bar must be a little less than any of the aper- tures. In all kinds of achromatic eye-pieces which are composed of single lenses, flint-glass should be employed, because it has the greatest refrac- tive power, and therefore requires a less curva- ture to have the same focal distance. The spher- ical aberration, consequently, which always in- creases with the curvature of the lenses, will be less in a flint-glass eye-piece, than in one of crown-glass. Flint-glass, indeed, produces a greater separation of colours, but the error aris- ing from this cause is completely removed by the proper arrangement of the lenses. Vol. H. 45O OPTICS. On Eye-Pieces with Four Lenses^ which remove the Chromatic Aberration. Eye-pieces ^ good achromatic eye-piece may be made of with four o . ill i r lenses. four lenses, if their rocal lengths, reckoning irom that next the object, be as the numbers 14, 21 , 27, 32, their distances 23, 44, 40, their aper tures 5.6 ; 3.4; 13.5; 2.6; and the aperture of the diaphragm placed in the anterior focus of the 4 th eye-glass, 7. In one of Ramsden's small telescopes, whose object glass was 8|- inches in focal length, and its magnifying power 15.4, the focal lengths of the eye-glasses were O.77 of an inch; 1.025; 1.01 ; 0.79, and their respective distances, reck- oning from that next the object, were 1.18; 1.83; 1.10. In the excellent telescope of Dollond's con- struction, which belonged to the due de Chaul- nes, 3 the focal length of the eye-glasses, begin- ning with that next the object, were 14^ lines, 19, 22|, 14, their distances 22.48 lines ; 46.17 ; 21.45; and their thickness at the centre 1.23 lines; 1.25 ; 1.47. The fourth lens was plano- convex, with the plane side to the eye, and the rest were double convex lenses. On Achromatic Eye-Pieces for Astronomical Telescopes. Achromatic eye-pieces j n eye-pieces of this 'kind which invert the ior astrono- ,. i r , , i r i /* i i i. micai tele- object, the focal length or the first lens should scope*. 3 See page 434. " ':..;' OPTICS. 451 be triple that of fhe second, and their distance double the focal length of the second, or y of the focal length of the first. The lenses should be plano-convex, the plane surfaces turned to the eye, in order that the aberration of sphericity may be diminished as much as possible. 4 The telescope of Dollond's, belonging to the due de Chaulnes, had two astronomical eye- pieces, one of which was furnished with a mi- crometer. In the eye-piece which carried the micrometer, the first lens was 12^ lines in focal length, and 1.62 lines thick; the second was 5.45 lines in focal length, and 1.25 thick, the distance between their interior surfaces 4.2O lines, and the distance of the first lens from the focus of the object glass 13| lines. In the other eye-piece the focal length of the first lens was S.30 lines, and its thickness 1.60 ; and the focal length of the second was 3.53, and its thickness O.97 lines. In both these eye-pieces the lenses were plano-convex, with the plane surfaces turn- ed to the eye. See page 444. Ff 2 OPTICS. ON THE CONSTRUCTION OF OPTICAL INSTRUMENTS, WITH TABLES OF THEIR APERTURES AND MAGNI- FYING POWERS, AND THE METHOD OF GRINDING THE LENSES AND SPECULA OF WHICH THEY ARE COMPOSED. On the Method of grinding and polishing Lenses. On grind- .HAVING fixed upon the proper aperture and focal distance of the lens, take a piece of Sheet copper, and strike a fine arch upon its surface, with a radius equal to the focal distance of the lens, if it is to be equally convex on both sides ; or with a radius equal to half that distance, if it is to be plano-convex, and let the length of this arch be a little greater than the given aperture. Remove Formation w ith a file that part of the copper which is with- gages. out tne circular arch, and a convex gfige will be formed. Strike another arch with the same ra- dius, and having removed that part of the cop- per which is within it, a concave gage will be obtained. Prepare two circular plates of brass, Formation about -^ of an inch thick, and half an inch great- ei the tools. er j n diameter than the breadth of the lens, and solder them upon a cylinder of lead of the same diameter, and about an inch high. These tools SPTICS. 453 are then to be fixed upon a turning lathe, and one of them turned into a portion of a concave sphere, so as to suit the convex gage ; and the other into a portion of a convex sphere, so as to answer the concave gage, After the surfaces of the brass plates are turned as accurately as possible, they must be ground upon one another alternately, with flour emery ; and when the two surfaces exactly coincide, the grinding tools will be ready for use. Procure a piece of glass whose dispersive Formation power is as small as possible, if the lens is not for achromatic instruments, and whose surfaces are parallel ; and by means of a pair of large scissars or pincers, cut it into a circular shape, so that its diameter may be a little greater than the required aperture of the lens. When the rough- ness is removed from its edges by a common grind-stone, 1 it is to be fixed with black pitch to a wooden handle of a smaller diameter than the glass, and about an inch high, so that the centre of the handle may exactly coincide with the centre of the glass. The glass being thus prepared, it is then M . od o{ to be ground with the fine emery upon the gnn concave tool, if it is to be convex, and upon the convex tool, if it is to be concave. To avoid circumlocution, we shall suppose that the lens is to be convex. The concave too! 3 therefore, which is to be used, must be firmly fixed to a * When the focal distance of the lens is to be short, the surface of the piece of glass should be ground upon a com- mon grindstone, so as to suit the gage as nearly as pos- sible ; and the plates of brass, before they are soldered on the lead, should be hammered as truly as they can be done into their proper form. By this means much labour will be saved both in turning and grinding. Ff 3 454? OPTICS. table or bench, and the glass wrought upon it with circular strokes, so that its centre may never go beyond the edges of the tool. For every 6 circular strokes, the glass should receive 2 or 3 cross ones along the diameter of the tool, and in different directions. When the glass has received its proper shape, and touches the tool in every point of its surface, which maj be easily known by inspection, the emery is to be wash- ed away, and finer kinds* successively substi- tuted in its room, till by the same alternation of circular and transverse strokes, all the scratches and asperities are removed from its surface. Af- ter the finest emery has been used, the rough- ness which remains may be taken away, and a slight polish superinduced by grinding the glass \vith pounded pumice-stone, in the same manner as before. While the operation of grinding is going on 9 the convex tool should, at the end of every five minutes, be wrought upon the con- cave one for a few seconds, in order to preserve the same curvature to the tools and the glass. When one side is finished off with the pumice- stone, the lens must be separated from its handle by inserting the point of a knife between it and the pitch, and giving it a gentle stroke. The pitch which remains upon the glass may be ro- 4 Emery of different degrees of fineness may be made in the following manner. Take five or six clean vessels, and having filled one of them with water, put into it a consider- able quantity of flour emery. Stir it well with a piece of wood, and after standing for 5 seconds, pour the water Jnto the second vessel. After it has stood about 12 seconds, ppur it out of this into a third vessel, and so on with the rest ; and at the bottom of each vessel will be found emery of different degrees of fineness, the coarsest being in the first vessel, and the finest in the last. OPTICS. 455 moved by rubbing it with a little oil, or spirits of wine ; and after the ground side of the glass is fixed upon the handle, the other surface is to be wrought and finished in the ground and manner. When the glass is thus brought into its proper Mode of form, the next and the most difficult part of theP oli3hin S operation Is to give it a fine polish. The best, though not the simplest, way, of doing this, is to cover the concave tool with a layer of pitch, hardened by the addition of a little rosin, 1 to the thickness of -^j of an inch. Then having taken a piece of thin writing paper, press it upon the surface of the pitch with the convex tool, and pull the paper quickly from the pitch before it has adhered .to it ; and if the surface of the pitch By means is marked everywhere with the lines of the paper, pltc ' it will be truly spherical, having coincided ex- actly with the surface of the convex tool. If any paper remains on the surface of the pitch, it may be removed by soap and water ; and if the marks of the paper should not appear on every part of it, the operation must be repeated till the polisher, or bed of pitch, is accurately spherical. The glass is then to be wrought on the polisher by circular and cross strokes, with the oxide of tin, called the flpwers of putty in the shops, or with the red oxide of iron, otherwise called col- cothar of vitriol, till it has received on both sides a complete polish. 3 The polishing will advance J As colcothar. of vitiiol is obtained by the decomposi- tion of martial vitiiol, it sometimes retains a portion of this salt. When this portion of martial vitriol is decom- posed by dissolution in water, the yellow ochre which re- sults penetrates the glass, forms an incrustation upon its surface, and gives it a dull and yellowish tinge, which ij rommumcated to the image which it forms. 456 OPTICS. slowly at first, but will proceed rapidly when the polisher becomes warm with friction. When it is nearly finished, no more putty or water should be put upon the polisher, which should be kept warm by breathing upon it ; and if the glass moves with difficulty, from its adhesion to the tool, it should be quickly removed, lest it spoil the surface of the pitch. When any par- ticles of dust or pitch insinuate themselves be- tween the glass and the polisher, which may be easily known from the very unpleasant manner of working, they should be carefully removed, by washing both the polisher and the glass, other- wise the lens will be scratched, and the bed of pitch materially injured. By means The operation of polishing may also be per- of cloth, formed by covering the layer of pitch with a piece of cloth, and giving it ar spherical form by pressing it with the convex tool when the pitch is warm. The glass is wrought as formerly, upon the surface of the cloth, with putty or col- cothar of vitriol, till, a sufficient polish is induced. By this mode the operation is slower, and the polish less perfect ; though it is best fitted for those who have but little experience, and would therefore be apt to injure the figure of the lens by polishing it on a bed of pitch. In this manner the small lenses of simple and compound microscopes, the eye glasses, and the object glasses, of telescopes, are to be ground. In grinding concave lenses, Mr. Imison 4 em- ploys leaden wheels with the same radius as the curvature of the lens, and with their circumfer- ences of the same convexity as the lens is to be concave. These spherical zones are fixed upon School of Arts, part ii, p. 145. OPTICS. 457 ft turning lathe, and the lens, which is held steadily in the hand, is ground upon them with emery, while they are revolving on the spindle of the lathe. In the same way convex lenses may be ground and polished, by fixing the con- cave tool upon the lathe ; but these methods, 1 however simple and expeditious they may be,* yofgrind ~ should never be adopted for forming the lenses a lathe, of optical instruments, where an accurate sphe- rical figure is indispensable. It is by the hand alone that we can perform with accuracy those circular and transverse strokes, the proper union of which is essential to the production of a sphe- rical surface. On the Method of Casting, Grinding, and Polish- ing the Specula of Reflecting Telescopes. The metals of reflecting telescopes are gener- Co ally composed of 32 parts of copper, and 15 of tion f the ] . . J . , , ,,. . r r r metal. grain tin, with the addition or two parts or arsenic, to render the composition more white and com- pact. The reverend Mr. Edwards found, from a variety of experiments, that if one part of brass, and one of silver, be added to the preced- ing composition, and only one part of arsenic used, a most excellent metal will be obtained, which is the whitest, hardest, and most reflec- tive, that he ever met with. The superiority of this composition, indeed, has been completely evinced by the excellence of Mr. Edwards' te- lescopes, which excel other reflectors in bright- ness and distinctness, and shew objects in their natural colours. But as metals of this compo- sition are extremely difficult to cast, as well as to grind and polish, it will be better for those who OPTICS. are inexperienced in the art, to employ the com* position first mentioned. Method of After the flasks of sand 5 are prepared, and a, cuKthig the mO uld made for the metal by means of a wood- n 'V" en or metallic pattern, so that its face may be downwards, and a few small holes left in the sand at its back, for the free egress of the in- cluded air ; melt the copper in a crucible by itself, and when it is reduced to a fluid state, fuse the tin in a . separate crucible, and mix it with the melted copper, by stirring them together with a wooden spatula. The proper quantity of powdered arsenic, wrapt up in a piece of paper, is then to be added, the operator retain- ing his breath till its noxious fumes are com- pletely dissipated ; and when the scoria is re- moved from the fluid mass, it is to be poured out as quickly as possible into the flasks. As soon as the metal is become solid, remove it from the sand into some hot ashes or coals, for the purpose of annealing it, and let it remain among them till they are completely cold. The ingate is then to be taken from the metal by means of a file, and the surface of the speculum must be ground upon a common grindstone, till all the imperfections and asperities are taken away. When Mr. Edwards- composition is em- ployed, the copper and tin should be melted ac- cording to the preceding directions, and, when mixed together, should be poured into cold wa- ter, which will separate the mass into a number of vsmall particles. These small pieces of metal are then tp be collected and put into the crucible, The finest sand which 1 have met with in this coun- try, is to be found at Roxburgh castle, in the neigUboui> 1 ' of Kelso. OPTICS. 459 along \vith the silver and brass j after they have been melted together in a separate crucible, the proper quantity of arsenic is to be added, and a little powdered rosin thrown into the fluid metal before it is poured into the flasks. When the metal is cast, and prepared by the Method of common grindstone for receiving its proper figure, fr j" n gtli the gages and grinding tools are to be formed in tools, &c. the same manner as for convex lenses, with this difference only, that the radius of the gages must always be double the focal length of the specu- lum. In addition to the convex and concave brass tools, which should be only a little broader than the metal itself, a convex elliptical tool of lead and tin should also be formed with the same radius, so that its transverse may be to its con- jugate diameter as 10 to 9, the latter being ex- actly equal to the diameter of the metal. On this tool the speculum is to be ground with flour emery, in the same manner as lenses, with cir- cular and cross strokes alternately, till its surface is freed from every imperfection, and ground to a spherical figure. It is then to be wrought with great circumspection, on the convex brass tool, with emery of different degrees of fineness, the concave tool being sometimes ground upon the convex one, to keep them all of the same radius, and when every scratch and appearance of rough- ness is removed from its surface, it will be fit for receiving the final polish. Before the spe- ij e d ot culum is brought to the polisher, it has been the htmcs ' practice to smooth it on a bed of hones, or a convex tool made of common blue hones. This additional tool, indeed, is absolutely necessary, when silver and brass enter into the composition of the metal, in order to remove that roughness which will always remain after the finest emery 46O OPTICS. has been used ; but when these metals are not ingredients in the speculum, there is no occasion for the bed of hones. Without the intervention of this tool I have finished several specula, and given them as exquisite a lustre as they could possibly have received. Mr. Edwards does not use any brass tools in his process, but transfers the metal from the elliptical leaden tool to the bed of hones. By this means the operation is tsimplified, but we doubt much if it is, in the least degree improved. As a bed of hones is more apt to change its form than a tool of brass, it is certainly of great consequence that the spe- culum should have as true a figure as possible before it is brought to the hones ; and we are persuaded, from experience, that this figure may be better communicated on a brass tool, which can always be kept at the same curvature by its corresponding tool, than on an elliptical block of lead. We are certain however, that when the speculum is required to be of a determinate focal length, this length will be obtained more pre- cisely with the brass tools than without them. But Mr. Edwards has observed, that these tools are not only unnecessary, but ' really detriment- al.* That Mr. Edwards found them unnecessary, we cannot doubt, from the excellence of the spe- cula, which he formed without their assistance ; but it seems inconceivable how the brass tools can be in the least degree detrimental. If the mirror is ground upon 20 different tools before it is brought to the bed of hones, it will receive from the last of these tools a certain figure, which it would have received even if it* had not been ground on any of the rest ; and it cannot be questioned, that a metal wrought upon a pair of brass tools, is equally, if not more, fit for the OPTICS. 461 bed of hones, than if it had been ground merely on a tool of lead. When the metal is ready for polishing, the Method of elliptical leaden tools is to be covered with pitch, 6 about -~ of an inch thick, and the polish- er formed in the same way as in the case of lenses, either with the concave brass tool, or with the metal itself. The colcothar of vitriol should then be triturated between two surfaces of glass, and a considerable quantity of it applied at first to the surface of the polisher. The speculum is then to be wrought in the usual way upon the polishing tool, till it has received a brilliant lustre, taking care to use no more of the colcothar, if it can be avoided, and only a small quantity of it, if it should be found necessary. When the metal moves stiffly on the polisher, and the colcothar assumes a dark muddy hue, the polish advances with great rapidity. The tool will then grow warm, and would probably stick to the specu- lum, if its motion were discontinued for a mo- ment. At this stage of the process, therefore, we must proceed with great caution, breathing continually on the polisher, till the friction is so great as to retard the motion of the speculum. When this happens, the metal is to be slipped oft" the tool at one side, cleaned with soft leather, and placed in a tube for the purpose of trying its performance ; and if the polishing has been conducted with care, it will be found to have a true parabolic figure. 6 In summer, or when the pitch is soft, it should be hardened by the addition of a little rosin ; and should al- ways be strained through a piece of linen, in order to free it from impurities, and rough particU*. 462 OPTICS* ON MICROSCOPES. On the Single Microscope. f n t h e fi rs t volume, we have described the method of forming small glass globules for the magnifiers of single microscopes ; 7 and have also explained the manner in which the enlarged pic- ture is formed upon the retina. When the lenses are made, either by fusion, or, which is by far the more accurate way, by grinding them on spherical tools, they are then to be fitted up for the purpose of examining minute objects* The method which Mr. Wilson has adopted in his pocket microscope, is very ingenious, though rather devoid of simplicity, as it obliges us to screw and unscrew the magnifiers, when we wish to view the object with a larger or a smaller power* The simplest and the most convenient method of mounting single microscopes, is to fix the lenses xi, a, , c, d, in a flat circular piece of brass, CD> which can be moved round / as a centre, by the action of the endless screw A B 9 upon the tooth- ed circumference of the circular plate. After the object has been viewed by some of the mag- nifiers, it may be examined successively with all the rest, by a few turns merely of the endless screw. i magni. In the first volume, Mr. Ferguson has already shewn how to find the magnifying power of single microscopes ; but in order to save the trouble of calculation, we have computed the* following new table of the magnifying power of 7 See vol. i, p. 260, note 8- OPTICS. 403 convex lenses, from 1 inch to -~ of an inch in focal length, upon the supposition that the near- est distance at which we see distinctly is seven inches. The first column contains the focal length of the convex lens in lOO ths of an inch* The second contains the number of times winch such a lens will magnify the diameters of objects. The third contains the number of times that the surface is magnified ; and the fourth the numbed of tirries that the cube of the object is magnified. A table of a similar kind, though upon a much smaller scale, has already been published by Mr. Barker ; but he supposes the nearest distance at which the eye can see distinctly, to be eight inches, which, I am confident from experience, is too large an estimate for the generality of eyes. When we consider, however, that the eye examines very minute objects at a less distance than it does objects of greater magnitude ; we will find that the magnifying power of lenses ought to be deduced from the distance at which the eye examines objects really microscopic* This circumstance has been overlooked by every writer 1 on optics, and merits our attentive consideration. I have now before me two specimens 6"f engrav- ing. The one is so large that I can easily read it at the distance of 1O inches. The other, which is a watch papef, beautifully engraven by Kirkwood and Sons, contains the Lord's pray- er in a circular space 7 of an inch in diameter, and is so exceedingly minute, that I cannot read it at a distance exceeding 5 inches. Now I main- tain, that if these two kinds of engraving are seen through the same microscope, the one will be twice as much magnified as the other. This indeed i$ obvious, for as the magnifying power of a lens i$ equal to the distance at which the object is ex- OPTICS. amined by the naked eye, divided by the focal distance of the lens, we shall have for the X number of times that the watch paper is magni- fied, and for the number of times that the X large engraving is magnified, x being the focal length of the lens. It follows, therefore, that the number of times that any lens magnifies objects really microscopic, should be determined by making the distance at which they are examined by the naked eye 5 inches. Upon this principle I have computed Table II, which contains the magnifying power of con- vex lenses, when employed to examine micro- scopic objects. OPTICS. 465 TABLE J. iViir TABLES oftlie Magnifying Power of small Convex Lenses, or Single Micros -opes, the distance at which the eye sees dis- tinctly being 7 inches. Focal distance of the lens or microscope. Number of times that the diameter bfrtjan object is magnified. Number of times that the surface o an object is mag nified. Number of times that the tube of an object is magni fied. ei new table for single ini- croscopci. ioo ski of an incn. ^p. Dec. of a Tunes ' Time. Timer Times. 1 100 7.00 49 343 i'.- 75 9.33 87 81O 1 5O 14.00 196 2744 40 17.50 306 536O -l- 30 23.33 544 12698 TV 20 35.OO 1225 42875 19 36.84 1354 49836 18 38.89 1513 58864 17 41.18 1697 69935 16 43.75 1910 83453 15 46.66 2181 1O1848 14 50.OO 2500 125OOO 13 53.85 2894 155721 12 58.33 3399 198156 11 63.67 4O45 257259 TV 10 70.00 ' 49OO 343COO 9 77.78 6O53 470911 8 87.50 7656 669922 7 100.00 1OOOO lOOOOOO 6 116.66 13689 16O16I3 5 140.OO 196OO 2744OOO TT * 175.00 3O625 5359375 3 233.33 54289 12649337 To 2 350.00 1225OO 428750OO 1 700.00 49OOOO 3430OOOOO" n. 466 OPTICS. TABLE II. A NEW TABLR of the Magnifying Power vf Small Convert Lenses, or Single Microscopes, the distance at s u u S O O CO *-O CO TJ< co GO co ^h -< 01 CM f-H *? I U U ; o jz HS-K . J D t^ ^O ^* I!** 0^ Ol ^f* ^O t^*" l-H _! I~*o3 a 'o c~~ci>" i2 u d d d d d 4 -s> a s S ^ |Sj is*1% 3 xj -is u S * -e u q l-H C^ ^Q lf^ J^^ ^ *Q CO 00 O d d d d -* O l-H i-* |gj|| '^ r-i * "> cS 2 *S * o * = O -H l-H l-H (N K fi< c *> ^l^oSl a | *. * J fi --4 u a 52^25 1! C j rt rt u fa 2 eg ^-5 -5 U yt 3 r* ^: -S o S t. . _ CO CO O ^ CO- J> CO C75 T}< |> f ^ S rt -^ M a S M s c " i ^ > U C t< i^ - ~ i CO CO I-H Qq co 5,> g S 1-5 a k * " ^ a a 00 *O ^ CM C5 Jh- CN co i> co ts 1 II -S S- *j c CO i i t~- C5 V5 . * -* CO CO R -S fl a * -^ a 11 O *o co I-H 11 >*5 I-H -O I/ "rt G "5 *S O 3 u -j R < O O *O CO CO ^ V> CO X i iiliil C d d d d d i Q n . 5 hoi "S 1 (5 J u d O O O CO *-* O 00 CO CN CN CN -2 Pal as >fl ft s r-l O o O O O CO CO H"5 CD ,C 3 |-S 3 1, JS s " ifj O >O CO O -H CO CO On the Cassegrainian Telescope. From the following table of the dimensions of *copc. Cassegrainian telescopes, founded on Dr. Smith's calculations, it appears, that though they are shorter by twice the focal distance of the small speculum than those of the Gregorian form with the same focal length, yet they have a greater magnifying power. A Cassegrainian telescope 15y inches in focal length, will magnify, accord- ing to the table, 93 times ; while a Gregorian one, with a similar speculum, magnifies only 86 times. This great difference between the per- formance of these instruments, does not exist merely in theory ; for Mr. Short constructed a telescope of the Cassegrainian form, of 24 inches focus, which, with an aperture of 6 inches, mag- nified 355 times. With this power, indeed, it was rather indistinct ; but it bore a power of 231 times with sufficient distinctness. In the Obser- vatory at Greenwich, there is a Gregorian tele- scope of Short's construction, which magnifies 25O times when the smallest mirror is employed, which is considerably less than the power of the Cassegrainian one of the same size. Reflecting telescopes are generally furnished with two or three small specula of different focal lengths, that the magnifying power may be va- ried without changing the eye-piece. I? " il 4 1 a I 'S *" 5 S CO CO CO 05 r^ *o CM J?jTcL I * -i CN .2 o ,r u S u c '" S 'o . ^ o - 1 C^ CO -H f - jf!| * 3 g * c O O O *^ O *o "H.'ou'S ^2 -H *"* a i _ C co u CO GO - j u o "5,- ^ 10 r- O .2 -> o u "3 2 -5 g O O ~ Q .a ~ J? S .c n 4) *5 t> " ft. O. a> ~~* 11111 I 4 -2 *- i/? *J CO u u q CO ^ ^ q q C^ CO ^* "^ *"* W ?< -! s g. 63 u u Q .c CD CD fa 2 ic b c E u ' w " *j | ^J ft> ^ *C OJ u ^_ u "S S T3 *- 3 S jj Q ^j M? r 1 S Z g |x"> , _c -^ -4 CN CD Q I J I -i. 1 ? " mi > IS 3 u u q r-. IO -" CO CM *O ^ i5^-S S- i CO >-O J3 i . iIs JB Z u V q t- r^ r- 1*1 u r-i CO CO **^ CM "S . u ^ . u "* *rd ^* "*^ *^ 5 ^ s JS O CD Jh- ^ W5 t- CO * [ f* Q^ C O J3 u d o d i *~ u Ji S , V ^ j j ^ ^_ ? 5 U i * fr q u= C *O *O CD CO CO O^ CM : K u 88 "5-s| ^ 5 L~ CD O r-H COCO fa *o 8* i 476 OPTICS. On the Newtonian Telescope. Newtonian] As the Newtonian telescope was powerfully recommended to the world by the simplicity of its construction, as well as by the name of its illustrious inventor, it is a matter of surprise that its merits should have been so long overlooked. During the last century Gregorian telescopes seem to have been universally preferred to those of the Newtonian form, till the celebrated Dr. Herschel introduced the latter into notice, by the splendour and extent of the discoveries which they enabled him to make. This philosopher, equally distinguished by his virtues and his talents, has constructed Newtonian telescopes from 7 to 4O feet 1 in focal length, by which he has great- ly enlarged our knowledge of the solar system, and disclosed many new and important facts re- I specting the structure of the heavens. PLATE i n the Newtonian telescope, the large para- Fig. 7.' bolic speculum is not perforated with a hole UV. A small elliptical plane mirror, inclined 45 de- grees to the axis of the tube, is placed at G H, about as much nearer the speculum than its fo- cus, as the centre of the small mirror is distant from the tube ; that is, the distance Gm of the small speculum from the focus of the great one, should be nearly equal to P T, half the diameter of the tube. The rays which form the image IK of the object AB instead of proceeding to form it at m, are intercepted by the plane spe- 1 A description of this noble instrument may be seen in the Phil. Trans. 1795, p l . 2. The diameter of the specu- lum is 4 feet, its thickness about 3 inches, and its greatest magnifying power 6000. *"** ; i*"** 1 .,,_ ,, _ r .. ,_^._ ,- . __.._.. i ,-mm ft- OPTICS. 477 culum at G H, and refracted upwards through an aperture in the side of the tube T T, where the image is formed and magnified by a double con- vex lens of a short focal distance. As the small plane mirror has an oblique po- Form of sition to the eye, it must be of an elliptical form. In order to find its conjugate or shortest diame- ter, say as the focal length of the great specu- lum is to its aperture, so is the distance of the small speculum from the focus of the great one to the conjugate diameter of the small mirror ; that is, the conjugate diameter df the ,, . Gm x DF T small mirror is rr 5 . Its transverse or P m f m V D F longest diameter will be ~ 5 x 1.4142: f m that is, equal to the conjugate diameter multi- plied by 1.4142 ; or, which is the same thing, its transverse will be to its conjugate diameter as 7 to 5, * which is nearly the ratio of the diagonal of a square to one of its sides. If a rectangular prism be substituted in place of the small mir- ror, having its sides perpendicular to the incident and emergent rays, the image will be erected, and a less quantity of light will be lost, than when the reflection is made from a mirror of the common kind. In most of Dr. Herschel's telescopes the plane Dr. H- mirror is hrown away, and the focal image /^ schel>sim - is viewed directly with a small eye glass, placed menu, at T E y the lower side of the tube. When the aperture of the speculum is very large, the loss of light occasioned by the interposition of part of 1 Mr. Adams in his Introduction to Nalural Philosophy, v. ii, p.. 534, erroneously observes, that the length of the small speculum should be to its breadth a 2 to 1. OPTICS. the observer's head is trivial ; but when the aper- ture is small, the speculum must be inclined a little to the incident rays. I have frequently taken a Newtonian speculum, 3^- inches in dia- meter, and 3O inches in focal length out of its tube, and viewed the moon in this manner with great satisfaction. The superior performance of Newtonian telescopes, without the plane mirror, can be conceived only by those who have made the experiment. f r ^ s lt 1S more difficult to find any of the heaven- Newtonian ly bodies with a Newtonian than with a Grego- tekscopes. r f an telescope, it has been customary to fix a small astronomical telescope on the tube of the former, so that the axis of the two instruments may be parallel. The aperture of its object glass is large, and cross hairs are fixed in the focus of the eye glass. The object is then found by this small telescope, which is called the finder ; and if the axis of the instruments are rightly adjusted, it will be seen also in the field of the large tele- scope. When the Newtonian telescope, how- ever, is large, and placed upon its lower end to view bodies in great altitudes, the finder can be of no use, from the difficulty of getting the eye to the eye piece. On this account I would pro- pose to bend the tube of the finder to a right angle, and place a plane mirror at the angular point, so as to throw the image to one side, or rather above the upper part of the tube, that the eye piece of the finder may be as near as possible to the eye piece of the telescope. If the latter of these plans be adopted, the angular point, where the plain mirror is fixed, should be placed as near as possible to the focal image, in order that only a small part of the finder may stand above the tubej for in this way the eye can be trans- OPTICS. 479 ferred with the greatest facility from the one eye piece to the other. The advantages of this con- PLATE struction will be understood from Figure 7,* m> X*lfiT 7 where T T is part of a Newtonian telescope, D ' the eye-piece, and ABC the finder. The image formed by the object glass Ais reflected upwards by the plain mirror , placed at an angle of 45 degrees with the axis of the tube, and the image is viewed by the eye, glass at C. Those who have been in the habit of using the Newtonian tele- scope with the common finder, will be sensible of the convenience resulting from this contri- vance. The only table, containing the apertures, mag- Reasons for nifying power, &c. of Newtonian telescopes, "^^"^ which has hitherto been published, was calcu-forNewto- lated by Dr. Smith, 4 from the middle aperture mac and power of Hadley's excellent Newtonian te- lescope, as a standard, the focal length of the great speculum being 5 feet 2y inches, its aper- ture 5 inches, and power 208. A speculum, however, 3 feet and 3 inches in focal length, was wrought, by Mr. Hauksbee, to so great per- fection, as to magnify 226 times.* It shewed the minute parts of the new moon very distinct- ly, as well as the belts of Jupiter, and the black list or division of Saturn's ring. For these ob- jects, it bore an aperture of 3f or 4 inches ; but in cloudy weather it shewed land objects most distinct, when the whole surface of the metal was exposed, which was 4-f inches in diameter. Since the method of grinding specula, and giving them a true parabolic figure, is much better un- 4 Optics, vol. i, p. 148. Dr. Smith's table was con- tinued from J/ to 24 feet, by Mr. Edwards. J Smith's Optics, vol. u. Remark, p. 79, col. 2, 480 OPTICS. derstood at present than it was in the time of Mr. Hauksbee, Newtonian telescopes may be made as perfect as this instrument of his construction. Upon it, as a standard, therefore, we have com- puted the following new table, on the suppo- sition, that reflecting telescopes, of different lengths, shew objects equally bright and distinct, when their linear apertures, and their linear am- plifications, or magnifying powers, are as the square square roots, or biquadratic roots, of the cubes of their focal lengths ; and consequently, when the focal distances of their eye glasses are as the square square roots of their lengths. The first column contains the focal length of tlbk f th *^ e g reat speculum in feet, and the second its li- near aperture in inches, and 100 ths of an inch. The third and fourth columns contain Sir Isaac Newton's numbers, by means of which the aper- tures of any kind of reflecting telescope may be readily computed. 6 The fifth column exhibits the focal length of the eye glasses in 1000 ths of an inch ; and the sixth contains the magnifying power of the instrument. * See Gregory's Optics, Appendix, p. 229, an published an account of his discovery in the ed by Mr. Transactions of the Royal Society. 1 They con- Gl sisted merely of a drop of water, taken up on the point of a pin, and placed in a small hole at D, ~ of an inch in diameter, in the piece of P * ATEXII? brass D E, about ~ of an inch thick. The hole lg ' * D is in the middle of a spherical cavity, about ^ of an inch in diameter, and a little deeper than half the thickness of the brass. On the opposite side of the brass is another spherical cavity, half as broad as the former, and so deep as to reduce the circumference of the small hole to a sharp edge. The water being placed in these cavities, Description will form a double convex lens, with unequal ^* wa convexities. The object, if it is solid, is fixed scope. upon the point C of the supporter A J5, and placed at its proper distance from the water lens, by the screw FG. When the object is fluid, it is placed in the hole A, but in such a manner as 1 Phil. Trans. No. 221, 223. See also Smith's Opticg, vol. ii, p. 394. Hh 2 48-i OPTIGS. not to be spherical ; and this hole is brought op* posite the fluid lens, by moving the extremity G of the screw into the slit G H. Description From this microscope of Mr. Grey's, the one of the new W j 1 i c j 1 we are now to describe is totally different. fluid mi- . . n , ' -VTT croscope. It is represented, as fitted up, in rlate XI, rig. PLATE^XI^^ and some of its parts, on a larger scale, in 4 'Fig. 3 and 4. A drop of very pure and viscid turpentine varnish is taken up by the point of a. piece of wood, and dropped at a, upon the pi^ce of thin and well polished glass a bed I; and dif- ferent quantities being taken up, and dropped, in a similar manner, at b, c, d, will form four or more plano-convex lenses of turpentine var- nish, which may be made of any focal length, by taking up a greater or a less quantity of the fluid. The lower surface of the glass abcdl, having been first smoked with a candle, the black pig- ment, immediately below the lenses , b, c, d, is then to be removed, so that no light may pass by their circumferences. The piece of glass, ale, is next to be perforated at /, and surround- ed with a toothed wheel CD, which can be moved round / as a centre, by the endless screw A B. The apparatus CDBA is placed in a circular case, which is represented by B H in Fig. 2, and part of it, on a larger scale, by CD in Fig. 4, and to its sides the screw A B is fas- tened, by means of the two arms m, n. This circular case is fixed to the horizontal arm R, by means of a brass pin, which passes through its upper and under surfaces, and through the hole /, (Fig. 3), --vhich does not embrace the pin very tightly, in order that CD may revolve with facility. On the upper surface of B H is an aperture AT, directly above the line described by the centres of the fluid lenses, when moving OPTICS. . 485 round 1 ; and in this aperture is inserted a small cap, with a little hole at its top, to which the eye ,is applied. E MN is the moveable stage, that carries the slider P ? on which microscopic ob- jects are laid ; and is brought nearer, or removed from, the lenses by the vertical screw DE. RS is the perpendicular arm to which the microscope is attached. FG is the pedestal j and C is a plain mirror, which has both a vertical and horizontal motion, in order to illuminate the objects on the ilider. When the microscope is thus constructed, the Method of object to be viewed is placed upon P, and the^J" 511 "^' icrew A B is turned, till one of the lenses be directly below the aperture K. The slider is then raised or depressed, by the screw D E, till the object be brought into the focus of the lens. In this manner, by turning the screw AB, and bringing all the lenses, one after another, direct- ly below A", the object may be successively exa- mined with a variety of magnifying powers. The focal lengths of these fluid lenses will in- crease a little after they are formed ; but if they are preserved from dust, they will last for a long time. The turpentine varnish should be as pure and viscid as possible j the glass on which it is dropped should be very thin; and the microscope should stand on a horizontal surface. I have even employed these fluid lenses as the Compound object glasses of compound microscopes ; and I j^ """ once constructed a compound microscope, in which both the object-glass and eye-glass were made of turpentine varnish. It performed much better than I expected, but rather gave a yellowish tinge to the objects which were presented to it, llli 3 486 OPTICS. ACCOUNT OF AN IMPROVEMENT ON THE CAMERA OBSCURA, AND OF A NEW PORTABLE ONE UPON A LARGE SCALE. JL HE camera obscura, which is one of the sim- plest and most amusing of our optical instruments, has already been described in the first volume. * The improvements which have been made upon it since its first invention, regard chiefly its ex- ternal form, and no attempts have been made to increase the brilliancy and perfection of the causes of image. When we compare the picture of ex- ^elllTthe ternal objects, which is formed in a dark cham- cameraob- ber by the object-glass of a common refract- ing telescope, with that which is formed by an achromatic object-glass, we will find the difference between their distinctness much less than we would have at first expected. Although the achromatic lens forms an image of the mi- nutest parts of the landscape, yet when this image is received on paper, these minute parts are obliterated by the small hairs and asperities on its surface, and the effect of the picture is very much impaired. In the Royal Observatory at Greenwich, the image is received upon a large concave piece of stucco, but from the testimony of those who have witnessed its effects, this sub- 1 See vol i, pp. 285, 286. OPTICS. 487 stance does not seem to be more favourable for the reception of images than a paper ground. In order to obviate these inconveniencies, I tried a number of white substances of different degrees of smoothness, and several metallic surfaces with different degrees of polish, but did not succeed in finding any surface superior to paper I hap- pened, however, to receive the image on the t hescand silvered back of a looking glass, and was surpris- rendering 1 Ml- 1 ! ' L. 1-* U l " e im a g e ed at the brilliancy and distinctness with wnicn more external objects were represented. The little ''ant and spherical protuberances, however, which arise from the roughness of the tin-foil, have a tenden- cy to detract from the precision of the image, and certainly injure it considerably when exa- mined narrowly with the eye. In order to re- move these small eminences, I ground the sur- face carefully with a bed of hones, which I had used for working the plane specula of Newto- nian telescopes. By this operation, which is exceedingly delicate, and may be performed without injuring the other side of the mirror, I obtained a surface finely adapted for the recep- tion of images. The minute parts of the land- scape are formed with so much precision, and the brilliancy of colouring is so uncommonly fine, as to equal, if not exceed the images which are formed in the air by means of concave spe- cula. Notwithstanding the blueish colour of the metallic ground, white objects are represented in their true colour, and the verdure of the foliage appears so rich and vivid, that the image seems to surpass in beauty even the object itself. On account of the metallic lustre of the surface, the distinctness of the image will always be greatest when the eye of the observer is placed in the di- rection of the reflected rays. 488 OPTICS. Decription The common portable camera obscura, which portawT nas already been described, 1 is necessarily on a camera ob- small scale, and is very far from being con- venient. These inconveniencies are completely remedied in the camera obscura invented by my friend the Rev. Mr. Thomson of Duddingston, PIATE which is represented in Figures 5 and 6 of Plate * lv ' XIV. In Fig. 5, A is a metallic or wooden ring, in which the four wooden bars A F, A 1, A G 9 A H, move by means of joints at A; and are kept asunder by the cross pieces B C 9 D , which move round B and D as centres, and fold up along BA 9 and D A, when the instrument is not used. The surface F 1 G 77, on which the image is received, consists of a piece of silk covered with paper. It is made to roll up at 7 77, which moves in a joint at 7, so that the whole surface F I H G, when winded upon IH, can be fold- ed upon the bar I A. By this means, the in- strument which is covered with green silk, lined with a black substance, may be put together and carried as an umbrella. It is shewn more fully i%. 6. in Fig. 6, where A is the aperture for placing the lens, and B C a semi-circular opening for viewing the image. A black veil may be fixed to the circumference of B C, and thrown over the head of the observer to prevent the admission of any extraneous light. See vol, i, p. 286. 480 DIALING. DESCRIPTION OF AN ANALEMMATIC DIAL, WHICH SETS ITSELF. THE analemmatic dial is represented by CD i Fig. 2 of Plate XII, and is generally described upon the same surface with a horizontal dial AB, Fig. for the purpose of ascertaining its proper posi- tion, without the assistance of a meridian line or compass. It is always of an elliptical form, ap- proaching to that of a circle, as the place for which it is made recedes from the equator. Its stile is perpendicular, and has different positions in the line 25 vj 1 , changing with the declination of the sun, and indicated by the names of the months marked upon its surface. From the ob- liquity of the stile of the one dial, and the rect- angular position of the other, the motion of their shadows is so different, that the dial may be reck- oned properly placed when the shadows of both stiles indicate the same hour. In order to understand the theory and con- Theory of struction of this dial, let BE be its length per- * c dial pendicular to the direction of the meridian. lg% ' Having bisected B E in A^ make A equal to the sine of the latitude of the place ; and with the cosine of the latitude as radius, set off AD and A C equal to the tangent of 23 28', the 49O DIALING. sun's greatest declination. The points D and C are the places of the stile in the time of the sol- stices, on the 21 st of June and December ; and if the tangent of the sun's declination for the first day of every month is set off in a similar man- ner between A and /), and A and C, the points thus found will be the place of the stile on those days, and the radius B C drawn from all these points to B will be the hour line of six at these different times. In order to prove this, let Z M N H (Fig. 5), be the meridian, P p the six o'clock hour circle, and P H the height of the pole, then AZ.8 is the azimuth of the sun, and P Z S its complement, AS the sun's decimation, and PS its comple- ment. Now, in the spherical triangle P Z S right angled at P, we have by spherical trigono- metry (Playfair's Euclid, prop. XVIII.) Radius : Sin. P Z=Tang. P Z V : Tang. P S, that is, Ra- dius: Sin. PZ Co Tang. Azimuth : Co Tang, declination, for P Z S is the complement of the azimuth, and P S the codeclination ; but as ra- dius is a mean proportional between the tangent and cotangent (Def. IX, Cor. 1 , plane trigonom.), the tangents will be in the reciprocal ratio of the cotangents, and consequently cotang. azi- muth : cotang. declin. Tang, declin. : Tang. azimuth. Therefore, Rad. : Sin PZ Tang, declin. : Tang. : azimuth ; and the sine of P Z the colatitude, is the same as the cosine of the latitude. Now, if A C represents the six o'clock hour line when the sun is in the equator, and A C the tangent of the sun's decimation, for a radius equal to the cosine of the latitude, or A C Tang, declin. X cosin. latitude, the angle ABC will be equal to the sun's azimuth, for from the last DIALING, 491 analogy, Tang, declin. x cos. latitude Rad. x Tang, azimuth, therefore A C Rad. X Tang, azimuth ; that is, A C is equal to the tangent of the sun's azimuth when AB is radius ; and con- sequently A B C is the sun's azimuth since A C is its tangent. If the sun were in the equator and the stile at A, his azimuth from the south would be A 7?, whereas when the stile is at .(?, his azimuth is CB, which is equal to A B A B C ; therefore A B C is the sun's azimuth from the east or west at six o'clock, and B C the six o'clock hour line. In the same way it might be shewn, when the stile is placed in any point between C and D, that a line drawn from it to the point B will be the six o'clock hour line for that declination, and that the angle at , com- prehended between this line and A B, will be equal to the azimuth of the sun. In order to determine the horary points and the circumference of the dial, we must consider, that if the equator be projected upon the horizon of any place, it will form an ellipse whose con- jugate or shortest diameter is equal to the sine of the latitude of that place. Let BMF there- fore, be the equator projected on the hori-rig.6: zon of a given place, so that A M half the con- jugate axis is to A B, half the transverse axis, as the sine of the latitude of that place is to ra- dius. Then having described the semicircle BXIlt\ divide the quadrants B Xll, and XII F, into six equal parts for the hours, into 12 for the half hours, and into 24 for the quarters, each hour being 15 degrees in the daily motion of the sun, each half hour 7 30', and each quarter 3 45', and from these points, from the point 7/7, for example, draw II 1C E parallel to A XI I, or perpendicular to A B, the point- 492 DIALING. C where this line cuts the ellipse will be the horary point, and D C will be the three o'clock hour line when the stile is at D. As there is some difficulty, however, in de- scribing an ellipse with accuracy, we shall shew how to find the horary points without describ- ing this conic section. Take B C equal to the breadth of the dial, and having bisected it in A 9 draw A 12 perpendicular to BC, and equal to the sine of the latitude, A C being radius. Then upon the centre A^ with the distance A 12, de- scribe the semicircle D 1 2 F 9 and with the dis- tance AB the semicircle CHE. Divide the quadrant H B into six equal parts for hours in the points m, ?z, 0, p, q, and the quadrant 12 R JRto the same number of equal parts in the points a, b 9 c, d, e; and through a, /', c, &c. draw a 1 1 , b 1O, c 9, &c. parallel to C B ; and through m } n, o, &c. draw m 1, n 2, o 3, parallel to HA ; the points of intersection 1 , 2, 3, 4, 5, will be the horary points, and will be in the circum- ference of an ellipse. The horary points being thus known, it is not necessary to traee the el- lipse, otherwise it might be easily done with the hand. If the divisions Hm 9 mn t &c. are sub- divided into half hours and quarters, or even lower, the corresponding points in the ellipse 1 2 B may be determined in a similar manner. In order to demonstrate that C is the horary point of three o'clock, and D C the hour line when the sun is at his greatest north declination, we must find from the construction the angle C D My or the sun's azimuth, reckoned from the south, and see if the triangle PZ S (Fig. 7) furnishes us with a similar expression of the angle Z, or sun's aziimith. In Fig. 6, C H 9 or its equal A E, is evidently the sine of the horary DIALING. angle, AB being radius ; and since CE or AH is the cosine of the horary angle, in a circle whose radius is AM, or the sine of the latitude, \ve will have C E or A //Cos. horary angle X Sin. lat. But according to the first part of the construction .///) Tan. declin. X Cos. lat. ; there- fore D H, the difference between A D and A H will be zz Cos. hor. angle x Sin. lat Tang, de- clin. x Cos. lat. ; and the tangent of the angle CH CD H or jjjj. will then be equal to _ Sin. Hor. dngle _ Cot. Hor. Angle X Sin. Latit. Tang. Bed. X Cos. Lat. Now, in order to find a similar expression for the angle P Z S, (Fig. 7) let SO be a perpen- Kg- r- dicular upon P Z : and the Sines of the seg- ments P 0, Z 9 will be reciprocally proportional to the angles at the base P and Z, (Flay fair's Spher. Trig. Prop. XXVII) ; that is, Sin. Z : Sin. P ir Tang. P : Tang. Z ; and therefore, rr r, Sin. PO X Tang. P -r, . o - ~ ~ Tanjr. Z zr - . ,. vn * . But, Sin. Z O zz t> vn ZO Sin. P'LPO*=Sm. PO x Cos. PZ Sin. PZ xCos. P 0. Now, since Rad. : Tang, zr Sin. : Cosine, and since Cos. : Sin. zz Rad. : Tang. we have, by the rule of proportion, Sin. P Ozz Cos. P X Tang. P ; and Tang. PO zz ^? Therefore S/a.PO, Co/. PVxTang.PO "' re> Sln.ZO~ Sin. PO X C*J. PZSin. PZ X Cox. f Dividing by Cos. P we have. Sin.PO_ _ Tang. PO _ , . Sin. ZQSin. PO X Cos. PZSin.PZ > '" Cot. PO * See Trail's Algebra, Appendix> No. VI, on the arith- tucl'i.c ofSinet, Theorem II. 494* DIALING. Tang. PO=~-pQ, we shall have, by substitution, &. P0 7. po _ Sin. ZQ Tang. PO X Cos. PZSin. PZ Again, by Playfair's Spher. Trigon. Prop. XXI, Cos. P : Rad. = Tang. P : Tang, P S, conse- quently Tang. P 0=Tang. P S X Cos. P. Sub- stituting, therefore, this new value of Tang. P in its room, in the last equation, multiplying the whole by Tang. P, and dividing by Tang. PS,* we shall have, Tang. P X Sin. P0_ Cos. P X Tang. P _ Sin. ZO ~~ Cos. PZ X Cos. PSin. PZ X Cot. PS ButsinceTang.:Rad. Cos.rSin.jSin.P Cos. P X Tang. P. By substituting Sin P in place of its value we shall have Tang. Z, or its equal, Tang. P X Sin. PO' Sin. P _ Sin. ZO Cos. P X Cos. PZSin. PZ X Cot. PS that is, by substituting the names of the symbols 'P rj _ Sin. Hor. single * Cos.Hor.Ang. X Sin.Lat. Tang. Dec. X Cos.Lat. which is the same expression of the tangent of the sun's azimuth, or angle Z, as was deduced from the former construction. its con- The analemmatic dial being thus demonstrated, struction. j ts construction will be better understood by tak- ing an example. Let it be required, therefore, to construct one of these dials for latitude 56 degrees north, which nearly answers to Edin- Fig. 3 . burgh. Let A C (Fig. 3) be taken for half the .* . breadth or radius of the dial, and let it be divid- ed into 1OOO parts, then A 12, which must be equal to the sine of the latitude, or 56 degrees, will be 829, which are the three first figures of the * Since the tangents are in the inverse ratio of the co- tangents, multiplying any number by the cotangent, is the same as dividing it by the tangent. DIALING. 495 natural sine of 56 degrees in a table of sines. In order to find the points D, C 9 (Fig. 4) where the F g-4- stile is to be placed at the solstices on the 21" of June and December, take the tangent of 23 28', the sun s decimation at that time, and it will be 434, if the radius were AC or 1OOO j but as the ra- dius is the cosine of the latitude, which is 55Q, we must say as 1OOO : 559434 : 243, the length of AD and AC. On the 21" of Feb- ruary, April, August, and October, the sun's de- clination is nearly 11 19', the tangent of which for a radius of 10OO is 200 , but for a radius of 559, the cosine of the latitude, it will be 112, which is the distance of the stile from A on both sides on the 21" of the months already mentioned. On the 21" of January, May, July, and November, the sun's declination is nearly 20 8' the tangent of which, for the radius 1 000, is 367 ; but for the radius 559 it will be 205, which is the distance of the stile from A^ on both sides, on the 21 st of these months, the names of the months being inserted beside the points, as in Fig. 2. The horary points are now to be de- termined in the manner already mentioned, 1 and the dial will be finished. In order to place the dial, we have only to turn it round till the stile of the analemmatic dial indicates the same hour with that of the horizontal one, and it will then be properly placed. Sec p. 491. 496 DIALING. DESCRIPTION OF A NEW DIAL IN WHICH THE HOURS ARE AT EQUAL DISTANCES IN THE CIRCUMFER- ENCE OF A CIRCLE.* W ITH any radius describe the circle FXII B: pi* 1 T XIT ^ raw 4-XI-I f r the meridian, and divide the tig. 6. 'quadrants FJKII, B XII, each into six equal parts for hours. To the latitude of the place add the half of its complement, or the height of the equator, and the sum will be the inclination of the stile, or the angle D AC. Thus, at Edin- burgh, the latitude is 55 58', the complement of which, or the altitude of the equator, is 34 2' ; the half of which is 17 1', being added to 55 58', gives 72 59' for the inclination of the Stile or the angle D AC. The position of the stile, in the figure is that which it must have on the 21 st of March and September, when the sun crosses the equator ; but when the sun has north declination, the point A must move to- wards Z), and when he is south of the equator, it must move in the opposite direction. In or- * This dial was invented by M. Lambert, and is de- scribed and demonstrated in the EphemerideS of Berlin, .1777* P 200, written in German* DIALING. 497 der to find the position of the point A for any declination of the sun, multiply together the ra- dius of the dial, the tangent of half the height of the equator at the place for which the dial is con- structed, and the tangent of the sun's declina- tion, and the product of these three quantities divided by the square of the radius of the tables, will give the distance of the moveable point A from the centre of the circle FXI1B. Let it be required, for example, to find the position of the point A on the 21" of December and June, when the declination of the sun is a maximum, or 23 28', the radius A B of the dial being divided into 1OO equal parts. Log. 10O 2.0OOOOOO Log. Tang. 17 1^9.4857907 J.og.Tang.23 28' 9.6376106 Sum 21.1 23401 SzrLog.ofproduct. From this logarithm subtract 20, the logarithm of the square of the radius, and the remainder will be 1.1234O13 Log. 13.29. Take 13j parts, therefore, in your compasses, and having set them both ways from A, the li- mits of the moveable stile will be marked out. For any other declination, the position of the point A may be found in a similar manner. It will be sufficient in general to determine it for the declination of the sun when he enters each sign, and place these positions on the dial, as represented in Fig. 2. The length of the stile AC 9 or its perpendi- cular height H C 9 must always be of such a size that its shadow may reach the hours in the cir- cle FXIIB. For any declination of the sun, its length A C may be determined by plain tri- gonometry. A XII is always given, the inclin- VoL If. I i 498 DIALING* ation of the stile D A C is also known, the angle AXHC is equal to the sun's meridian altitude, and therefore the whole triangle may be easily found in the common way, or by the following trigonometrical formula : A C the length of , ., AXII* Sin. Merid. Alt. G ~ Sin. (180 Angle of Stile + Merid. Alt.) improve- Notwithstanding the simplicity in the con- iTb ' La n struct i n of tm>s dia lj tne motion of the stile is Grange, troublesome, and should if possible be avoided. For this purpose the idea first suggested by the celebrated La Grange will be of essential uti- lity. He allows the stile to be fixed in the centre A) and describes with the radius AE^ circles upon the different points where the stile is to be placed between A and Z>, and on the other side of AI which is not marked in the figure. All these circles must be divided equally into hours like the circle F XII B^ and when the sun is in the summer solstice, the divisions on the cir- cle nearest the stile are to be used ; when he is in the winter solstice, the circle farthest from A must be employed, and the intermediate circles must be used when the sun is in the intermediate points. This advice of La Grange may be adopt- ed also in analemmatic dials. 499 ASTRONOMT. ON THE CAUSE OF THE TIDES ON THE SIDE OF THE EARTH OPPOSITE TO THE MOON. IT has always been reckoned difficult for those onthe unacquainted with physical astronomy, to un- cause of thc derstand why the sea ebbs and flows on the side site thc P * of the globe opposite to the moon. This fact, moon, indeed, has frequently been regarded, and some- times adduced, by the ignorant, as an unsur- mountable objection to the Newtonian theory of the tides, in which the rise of the waters is referred to the attraction of the sun and moon. From an anxiety to give a popular explanation of this subject, Mr. Ferguson has been led into an error of considerable importance, in so far as he ascribes the tides on the side of the earth oppo- site the moon, to the excess of the centrifugal force above the earth's attraction. 1 It cannot be questioned, indeed, that the earth revolves round the common centre of gravity of the earth and moon, at the distance of nearly 600O miles from that centre ; and that the side of the earth op- posite the moon has a greater velocity, and con- Sec vol. i, p. 48, 49. li 2 500 ASTRONOMY. sequently a greater centrifugal force than the side next the moon ; but as the side of the earth farthest from the moon, is only 1O,000 miles from the centre of gravity, it will describe an orbit of 31,415 miles in the space of 27 days 8 hours, or 656 hours, which gives only a velocity of 47 miles an hour, which is too small to create a centrifugal force, capable of raising the waters of the ocean. PLATE v, The true cause of the rise of the sea may be Hg-*- v/ understood from Plate V] Fig. 4, where ABC is the earth, the common centre of gravity of the earth and moon, round which the earth will revolve in the same manner as if it were acted upon by another body placed in that centre. Let A MI B N 9 C P, be the directions in which the points AI .5, C, would move, if not acted upon by the central body ; and let B b n be the orbit into which the centre B of the earth is deflected from its tangential direction B JV. Then since the waters at A are acted upon by a force, as much less than that which influences the centre of the earth, as the square -B is less than the square of A, they cannot possibly be deflected as much from their tangential direction AM, as the centre B of the earth ; that is, instead of de- scribing the orbit A m, they will describe the or- bit e a. In the same manner the waters at c be- ing acted upon by a force as much greater than that which influences the centre B of the earth, as the square of B exceeds the square of C, will be deflected farther from their tangential di- rection than the centre of the earth, and instead of describing the orbit c p, will describe the or- bit h c i. As the earth, therefore, when revolving round the centre of gravity 0, will be acted upon bv 501 the m6on, in the same way as by another body placed in that centre, it will assume an oblate spheroidal form a b c ; so that the waters at c will rise towards the moon, and the waters at a will be left behind^ or will be less deflected than the other parts of the earth j 'by the lunar attraction^ from that rectilineal direction in which all re- volving bodies, if influenced only by a projectile force, would naturally mom li 3 IND;EX TO VOLUME SECOND. A ABSOLUTE fall of water, app. Page H4 Achromatic telescope, app. ;. 4 2 3 , object glass, double, app. - 428 tables of their radii, app. 436 . triple, 428 tables of their radii, app. 43 1 telescopes with four lenses, 438 formulae for, app. 440 eye pieces, - - 444 formulae for, app. 445> 447 Altitude of the sun, to find it by trigonometry, 54 Amontons on friction, app. 335 Analemmatic dial, app. 488 Anemometer, Leslie's, app. 278 Archimedes's screw engine, 113 ; mode of its operation, note, 114 Azimuth of the sun, to find it by trigonometry, 54, 59 B Balance, improvement upon it, app. - 385 Barker's water mill, without wheel or trundle, 97 various improvements upon it, app. 205 rules for constructing it, qpp. - 208 Besant's undershot wheel, app. - - 203 Bevelled wheels, on the formation of their teeth? app. 230 Blakey's hydraulic engine, 109 Blowing machine, app. "'. 415 Breast mills, app. - - 1 88 Bulfiager on friction, app. 335 Camera obscura, improvement upon it, app. 486 new portable, app. 487 Capstane, description of a simple and powerful one, app. 381 Carriage wheels, app. . '., 295 < conical rims disadvantageous in, app. 309 504 INDEX. Carriage wheels, inclination of spokes disadvantage- ous, app. - v - Page 302 Cassegrainian telescope, app. - V" 474 table for, app. 475^ Centre of gravity, mechanical method of finding it, app. 387 Clocks and watches, how to regulate them, - 66 Coulomb on wind mills, app. - - 264 . on friction, app. - - 339 Course of discharge, app. - .. -.. - 142 i of impulsion, app. - - ;' J -~ ' ib. Crane, description of a new and safe one, invented by the author, ^' +1^ - 89 Crown wheels, app. - - - 233 Cylindrical *un-dial for shewing the time of the day, the sun's place, and altitude, - 122 . method of using it, 1 25 D Declining dials, - - 4 Dial, horizontal, - - 3, 6, 16 . vertical, - - - 4, 6, 9 ' inclining, ... -4 reclining, -,- *< * *k. declining, ... ib. erect, direct, south, - 6, 9 erect declining, . - 10 . on a card, ... 19 i Pardie's universal, - - 22 .. double horizontal, - - v *; ' 6 1 Babylonian, i ' - ^ - /'$-. fl^ . Italian, '.''.' '.-'v- J D> .I. - Analemmatic, app. '':'''. - 488 Lambert's, description of, app. 495 Dials, method of placing them, ' V - 6$ to make three on three different planes with one gnomon, - - - - 126 universal, on a plain cross, - - 127 . by a terrestrial globe, and the shadows of several gnomons at the same time, 130 Dialing-lines, how constructed, - 17 by trigonometry, ... 4.2, .. principles and art of, /U*:I-T i;;vk> i by the terrestrial globe, 5 * cylinder, universal, h;^ui !3 .**V >ri 1 1 8 Dollond's Achromatic eye piece, afp. >ia* : 450 Double microscopes, app. - - 467 mills, app. - - 184 INDEX., 503 Eclipses, ; - Page 77, 79 limits of, . 8 period of, - ib. Engine, steam, app. - 389 Epicycloids, exterior, method of forming them me- chanically, app. - tf"'_ 234 geometrically, app. 236 . interior, app. - - -234 Equation of time, - "35 Erect, direct, south dial, - C, 9 declining dial, - - IO Euler on wind mills, app. - - 272 i on friction, app. 33 (J Finder, a new one for Newtonian telescopes, app. 478 Float-boards, number of, app. - - 147 . . size of, app. - - ib. - -- position of, app. - - 153 Flour mills, improvement upon, upp. - 203 Fluid microscopes, app. 483 - - Grey's, app; - - ib. -- < single, invented by the editor, app. 484 - double, Fluid object glass, Dr. Blair's, app. 443 Fly wheels on the nature and operation of, app. 353 - how they become regulators of machinery, app. - ib. ' how they become accumulators of power, app. 357 Friction, how to find the momentum of, in wind mills, a/>p. - r *> Page 8 i Pivots, form and size of, app. - .-;,r.i.-j i 162 Precepts for calculating the mean time of new and full moons, - * - *. 71 Pump mill, quadruple, 115 Pyrometer, description of an accurate one, 94 Quadruple pump mill, - "Vjfo 115 R Hackwork, method of forming its teeth, app. - 241 Rain wind, cause of it, app. - 417 Ram, hydraulic, app. - - - 420 Ramsden's achromatic eye piece, app. - 450 Reciprocating springs, - 106 Reclining dials, 4 Reflecting telescopes, app. - 472 table for, 473,475,481 Refracting telescopes, app. - 468 table for, - 47 1 Relative fall of water, app. - 144 Saw mill, note, - * ' 135 Screw engine, Archimedes's, * 113 Single microscopes, app. - - 462 Smeaton's maxims on undershot mills, app. - 286 Smeaton on windmills, app. - - 284 Specula, method of casting, grinding, and polishing, app. - *'.; - 457 Springs, reciprocating, ''- ro " IQ 6 Laywell, note, - "?'.' ib. Spur wheel, formation of, app. - - Steam engine, app. "T" Watt's app. - *' : * i(n 39 * Woolff's improvements on, app. 407 . on the power of, and the method of com- puting it, app. - - 4 1 1 Tables of the sun's place and declination - ^ 26 for calculating new and full moons and eclipses, 85 1 S D E X. 509 Tables of the equation of time, Page 32 Table of mean lunations, 84 M of the number of floatboards in undershot wa- ter wheels, according to Pitot, app. 149 _ for millwrights, constructed on new principles, app, - i?4> >7S , for breast mills, app. - - , - 190 for overshot mills, app. - - 201 of the velocity of thrashing machines, app. 326, 329 of the power of thrashing machines, app. 333 Tables for achromatic telescopes, app. 436 for single microscopes, app. - 46^ for refracting telescopes, app. - 47 j for Gregorian telescopes, app. - '473 for Cassegrainian telescopes, app. - 475 for Newtonian telescopes, app. 48 1 Teeth of wheels, method of forming them, app. - 210 , method of disposing them, app. - 234 Teeth of rackwork, method of forming them, app. 241 Telescopes achromatic, app. 423 refracting, app. 468 table for, app. . 47 1 1 Gregorian, app. - - . 472 table for, app. - 473 Cassegrainian, app. - 474 table for, app. - - 475 Newtonian, app. - - 476 table for, app. - 481 Herschel's, app. . 477, 482 Dollond's, app. . . _ 4.34 with fluid object glasses, app. 443 Thales's eclipse, . - . 79 Thrashing machines, app. ' -V . - 321 driven by water, app. - 32^ ~ driven by horses, app. 327 ; on the power of, app. 332 Tides, on the cause of, app. - ' -7 - 497 Traction, on the line of, aty. : ". ' 313 Trundle, , formation of, app. " *- . , 154 Turpentine varnish microscope, app. 4^4 U Universal dialing cylinder, ^ *\' Ix g Undershot water wheels, construction of, app. - 1 39 ~ table of the number of float- boards in, according to Pilot, app. >;-. .. - 149 Undershot mills, on the performance of them, app. 163, 134 510 s INDEX. V Vernier scale, nofe, - '''." ' Page 52 Verrier's windmill, description of, app. - 258 Vertical dials, - - - 4 Vince on friction, app. - - 338 W Water, how to measure its velocity, app. - 177 Water blowing machine, app. - - 415 Water microscopes, app. - 483 Water wheel, size of, app. - 147 number of floatboards in, app. - ib. position of floatboards, app. - ib. Wheels, bevelled, on the formation of their teeth, app. 236 Wheels, on the teeth of, app. - 210 Wheels, friction, app. 348 fly, app. - - ' 353 Wheels of carriages, on the advantages of large ones, app. - 298 on the position of, app. 3 1 2 Wheel carriages, app. - - 295 on the formation of, app. - 296 ' ' how to place the centre of gravity, app. - 317 Whitehurst's machine for raising water, app. - 419 Wind, method of finding its velocity, app. ' -_ 276 by Coulomb, app. 279 Windmill, description of Verrier's, app. 258 " those in Holland, app. , - 265 sails, on the formation and position of^ app. 266 angle of their inclination, according to Parent, app. 267 Euler's theorem concerning, app. - 268 Euler's observations upon, app. - 272 Windmills, table of their power, &c. app. 274 horizontal, app. " - 28* comparison between vertical and horizontal ones, app. - - 289 Smeaton's observations on, app. - 284 Coulomb's . app. - 265 Wipers of stampers, &c. mode of forming them, afp. 241 Printed by Munddl, Deig, and Stevenson, Edinburgh. d, ty .i-.i.h'jtf v /.. Mundell, Doig, &? Stevenson, Edinburgh, and Longman, Hurst, Rees, $ Orme; and T. Ostell, London. I. AN INQUIRY into the NATURE and CAUSES of the WEALTH of NATIONS, by ADAM SMITH, F.R.S. LL.D. with a LIFE of the Author, 3 vols 8 VO , One Guinea in boards. ADVERTISEMENT TO THE PRESENT EDITION' OF THIS WORK. For this Edition an Account of the Life of the Author has Seen drawn up ; and although it cannot be said that any Facts relating to that truly great Man are given, in addition to those which have already appeared, yet a more satisfactory Account, it is presumed, will now be found of his Studies and Doctrines, than has been prefixed to any other Edition of the Inquiry into the nature and causes of the Wealth of Nations. There are likewise prefixed, a Comparative View of the Doctrines of Smith and the French Economists, and a Method of facilitating the Study of Mr. Smith's Inquiry, by German Gar- nier, of the National Institute, Translator of this Work into the French language. The Advantage of some Directions to the Readers of this immortal Work, as it has justly been oalled, particularly to those who have not previously made the Science of Political Economy their Study, has been generally acknowledged. The following Observations are extracted from a Review which ap- peared of M. Garnier's Translation. ' Mr Gamier, in order to facilitate the understanding of his author, has laid down the heads of his work in the order in which he conceives they ' ought to have been treated ; and no doubt, had the course now sketched ' been followed by Dr. Smith, his book would have been read with more 1 pleasure and interest, and his doctrines would have been more easily ap- ' prehcnded. We 1 are of opinion, therefore, that the arrangement here 1 given, or iomethiBg on the same plan, might be advantageously prefixed ' t a future edition of the original.' Aff. to Monthly Rtvim, 1801. II. The FABLE of the BEES ; or, PRIVATE VICES, PUBLIC BENEFITS. With an ESSAY on CHARITY and CHARITY SCHOOLS, and a Search in- to the NATURE of SOCIETY. Also a Vindication of the Book from the Aspersions contained in a Present- ment of the Grand Jury of Middlesex, and an abu- sive Letter to Lord C . x " It was Dr. Mandeville who seems to have been at the bot- tom of all that has been written on this subject, (National Eco- nomy) either by Dr. Smith or the French Economists : That national wealth consists in industry, excited by necessity, na- tural or luxurious ; that the value and perfection of all the sub- jects of industry depend chiefly on the division of labour ; that certain labours or employments are productive, and others un- productive ; that it is mechanics or ploughmen that are demand- ed for national wealth, not men addicted to books, who often tend to make the poorer classes idle, vain, and discpntented ; that the value of articles depends on their scarcity and plenty. These are the leading principles in Dr. Mandeville's Fable of the Bees. Let any man of candour and common understanding peruse the Fable of the Bees, and the innumerable publications of the Economists, and then say, whether it be not almost certain, that the way was prepared for the inquiries and conclusions of the latter by those of the former. The introduction of the Fable of the Bees into France coincides with the time when the Eco- nomists received the impressions of education. As the just mode of investigation in natural philosophy was invented by English- men, so also the just mode of investigation in political economy, how to make a people powerful and happy, the most important of all the subjects of reasoning, was also first pointed out by an Englishman. Bacon and Newton were the fathers of legitimate inquiry in natural, and Mandeville in political philosophy. It is not a little astonishing, that the honour due, on this score, to Mandeville, has not been reclaimed before by his countrymen. Antijacob'm Review, Dec. 1805. Cdlegi Ubran University of California SOUTHERN REGIONAL LIBRARY FACILITY 305 De Neve Drive - Parking Lot 17 Box 951388 LOS ANGELES, CALIFORNIA 90095-1388 Return this material to the library from which it was borrowed. URL Form L9- THE UNIVERSITY LOS LIBRARY- ' fFORN 3 1158 00688 4166 _ 'ii mil mil i 1 i 'ii i AA 000081 515 9