TRANS®&CTIONS OF THE ROYAL IRISH ACADEMY. VOL. VIL RE 5 GR OF TRANSACTIONS OP HE ROYAL IRISH ACADEMY. VOL. VIL De ve ee er ee ee GEORGE BONHAM, PRINTER TO THE ROYAL IRISH ACADEMY. ———as 1800, fa REO Bao SAR LA TEA ORES ee ee ae THE ACADEMY defire it to be underfiood, that, as a body, they are not anfwerable for any opinion, reprefentation of fads, or train of reafoning, which may appear in the following papers. The authors of the Jeveral effays are alone refponfible for their contents. COO" N. EON sa SS, 1, ON the Preceffion of the Equinoxes. By the Rev. Matthew ai D.D. 8. F.T.C.D. and M.R.1. A. 5 Page 3 II. General Demonftrations of the Theorems for the Sines and Cofines of Multiple Circular Arcs, and alfo of the Theorems for expreffing the Powers of Sines and Cofines by the Sines and Cofines of Multiple Arcs; to which is added a Theorem by help whereof the fame Method may be applied to demonftrate the Properties of Multiple Hyperbolic Areas. By the Reverend Fobn Bauleda cd 4. M. shies bg fr: of Aficomny, and M.R.I. III. Remarks on the Velocity with which Fluids iffue from Apertures in the Veffels which contain them. By the Rev. Matthew Young, * D.D. 8.F.T.C. D.jand M.R. 1. As =e ; - 53 IV. A new Method of refolving Cubic a ‘ey Tho. Meredith, A, B. Trinity College, Dublin - * 69 V. On the Force of Teftimony in efabli ihing Fads co rary to daisy. By the Rev. Matthew Y ee .D.S8.F.T.C. D. and _M.R o Xe I A. - : - - ry > 9 9 VI. On the Number of the Primitive Colorific Rays in Solar Lights By the Rev. Matthew Young, D.D. S.F.T.C.D. and M.R.LA.- 119 VII. Obfervations on the Theory of Eleétric Attrattion and Repulfions . 74 the Rev. panty Miller, D.D. F.T,. C.D. and M. RA - 139 : VIII. 4 general COUR 7a a ee VIII. 4 general Demonftration of the Property of the Circle difcovered by Mr. Cotes, deduced from the Circle only. By the Reverend Fobn Brinkley, ras M. re ole effor of siete and M.R.1 A. Page 151 IX. Additional Obfervations on the Proportion of Real Acid in the Three Antient known Mineral Acids, and on the Ingredients in various Neutral Salts and other Compounds. By Richard Kirwan, E/q. L.L.D. F.R.S. and M.R.I.A. - - “sa eg X. Effay on Human Liberty. By Bieta cae Ele. L.L. 2 E.R.S. and M. R.1. A. 395 XI. Synoptical View of the State of the Weather at Dublin in the Year1798. By Richard Kirwan, Efy. L.L.D. F.R.S. and M.R.1.A. - 316 XII. An Abjtra® of Obfervations of the Weather of 1798, mede by » Henry Edgeworth, Efg; at sip cahts lit- in the 7 of Long ford in Ireland - 317 XIII. A Method of expreffing, when poffible, the Value of one variable Quantity in integral Powers of another and conftant Quantities, having given Equations expreffing the Relation of thefe variable Quantities. In which is contained the general Dottrine of Reverfion of Scries, of approximating to the Roots of Equations, and of the Solution of fluxional Equations by Series. By the RewsFohn Brinkley, A. M. Andrews’ Profeffor of Aftronomy, and M.R.I. A. - 325 XIV. Account of the Weather at Londonderry in the ee 1799+ By William Paterfon, M.D. and M.R.I. A. 357 > XV. Synoptical View of the State of the Weather at Dublin in the Year 1799. By Richard Kirwan, E/q. L:L.D. Pref. R.LA.and F.R.S. - 359 Lao SS ‘See PUOJLE, TE vile Re A bs UR hs XVI. Some Obfervations upon the Greek Accents. By Antler Bete ie Senior Fellow of Trinity College, Dublin 359 Pete toh Ne CU «FE. er a On the PRECESSION of the EQUINOXES, By the Rev. MATTHEW YOUNG, D. D.8.F. 7.6. D. & MRL A. Ir is univerfally acknowledged, that Sir Ifaac Newton has Read April, fallen into fome error in his calculation of the fun’s force ‘to ee produce the preceflion of the equinoxes, making it by one half lefs than the truth: but the particular fource of this’ error has not been fo generally agreed upon. Tuoven feveral excellent mathematicians, of whom D’Alam- bert feems to have been the firft, have given genuine fo- lutions of this problem, by. procefles entirely different from each other, perhaps it ftill may be worth while to endeavour to difcover diftin@ly in what confifts the fallacy of New'on’s reafoning, and whether in iome of the folutions of this curious gueftion, which are received as genuine, there do not lie fome fecret and unobferved errors, which being equal and contrary, compenfate each other, and thus leave the re‘ult correct, though the premifes from which it is deduced are faulty. A 2 THE liga Se Tus firft Lemma which Newton premifes to the inveftigation of the preceffion is as follows: «“ Ip A PE P_reprefent the earth, of uniform denfity, de- {cribed with the centre C, poles P, #, and equator AE; and if with the centre C and radius P C, the {phere Page he fup- pofed to be defcribed ; and QR be a plane perpendicular to the right line joining the centres of the fun and earth; and every particle of all the exterior earth Pap A Pe, which is’ higher than the infcribed fphere, endeavour to recede on either fide from the plane QR, and the effort of each particle be proportional to its diftance from the plane; I fay, firft, that the whole force and efficacy of all the particles in the circle of the equator A E, difpofed uniformly without the {phere, throughout the whole circumference, in the form of a’ ring, to turn the earth round its centre, is to the whole force and eflicacy of as many particles placed at the point A of the equator which-is moft remote from the plane QR, to move the earth round its centre with a like circular motion, as one to two. And that circular motion will be performed round an axis lying in the common interfection of the equator and the plane QR.” Tur demonftration of this Lemma is given in the Principia, and allowed to be legitimate. ral ra ad . a n o . es | His fecond Lemma is as follows : “ Tue fame things being fuppofed, I fay, fecondly, that the whole force and efficacy of all the particles without the {phere to turn the earth round its axis, is to the whole force of as many particles difpofed uniformly in the form of a ring, in the circumference of the circle AE of the equator, to move the earth, with a like circular motion, as two to five.” Tuz demonftration of this Lemma is alfo given in the Principles, and is likewife received as unexceptionable. s Lemma 3. “ Tue fame things being fuppofed, I fay, thirdly, that the | motion of the earth round the axis already defcribed, com- pounded of the motion of all its particles, will be to the motion of the aforefaid ring round the fame axis in a ratio, which is compounded of the ratio of the quantity of matter in the earth to the quantity of matter in the ring, and of the ratio of three fquares of the arch of a quadrant of a circle to two fquares of the diameter; that is, in a ratio of matter to matter, and of the number 925275 to the number 1000000.” Turs Lemma I fhall firft demonftrate in Newton’s fenfe, and then correét-the conclufion on the principles propofed by Simpfon ~ and Frifi. ' By ee) By the revolution of the circle EA H C, and citcumfcribed {quare (fig. 2.) PQS T round the common axis E H, let there be defcribed a fphere and circumfcribed cylinder. Let the radius AO. be = 1, the periphery of the circle AECH = p, the ordinate BR = y, abfcifla BO = x. Then: p::_x: px, the periphery of the circle whofe radius is OB; therefore px x2y will be the furface generated by the ordinate R G, in the revolution of the circle A E C H round the dia- meter EF H: but» will be the meafure of the velocity of the point B, therefore 2 / x? _y will be the momentum of all the particles in that furface; and the fluent of the quantity 2 px? y « will be the momentum of the entire fphere, when x is equal to the radius A O. But y= 1—x?/?; therefore the fluxion : ice rar CN eat.) ieee x2 xy = x? KX l—wrtice 5 SSS and the fluent I—wx? I —x7}5 ie oe of Salar 2), aan 2 X Circular arc ER —i« x 1—vx?\!, and the x4ox fluent of SSeS T3Xs 3 xcirculararcER 2.02 2 a: aX xx 1—x2s 8 therefore the whole fluent, when x =1, is 4 X quadrantal arc EA = 4.p; and 2p47 x” x 1—x7/i = 1. p*, the motion of the entire {phere. In a cylinder, the ordinate _» becomes = BR = 1; therefore the fluxion of the momentum of the cylinder = 2 p x? x, whofe fluent, when x=1,is 24. Therefore the motion of a cylinder is to the motion ae ae motion of an infcribed fphere, revolving round the fame fixed axis, and with the fame angular velocity, as 2 pto 7 £7, or as 16 to if, thatis, as four equal fquares to three circles infcribed in them. _ Ler the quantity of matter in an indefinitely flender ring, fur- rounding the fphere and cylinder at their common contact A OC, be reprefented by the letter m, its velocity willbe as AO = 1; and its motion = m, and therefore the motion of the cylinder is to the motion of the ring as ftom, or as 2p to 3m. Tue motion of the annulus, uniformly continued round the axis of the cylinder, is to its motion revolving uniformly in the fame pe- riodic time round one of its diameters, as the circumference of a circle to twice the diameter. For (fig. 2) let A R =z, and let its fluxion z be given, RB =), AB=x, and AO =7; let the motion be performed round the diameter A C, the velocity of the point R will be as R B or y; there- fore the fluxion of the motion of the annulus round the diameter AC, is to the fluxion of the motion round the center O in an immoveable plane, as z y to zr, that is, from the nature of a circle, as x toz; and therefore the motions themfelves are to each other in the fame ratio, that is, when x= AC, as the diameter to half the cir- cumference, or as twice the diameter to the circumference of a circle. Hence, by compounding all thefe ratios, the truth of the Lemma is manifeft. But C87 Bur Simpfon in his mifcellaneous tracts has juftly obferved, that though this reafoning be indifputably true in Newton’s fenfe, yet there is a difference between the quantity of motion fo confidered, and the momentum, whereby a body. revolving round an axis, endeavours to per’evere in its prefent ftate of motion, in oppofition to any new force imprefled, which latter kind of momentum it is that ought to be regarded in comput- ing the alteration of the body’s motion in confequence of fuch force. In this cafe, every particle is to be confidered as acting by a lever terminating in the axis of motion, fo that to have the whole momentum, the moving force of fuch par- ticle muft be multiplied into the length of the lever by which it is fuppofed to act; whence the momentum of each particle will be proportional to the fquare of the diftance from the axis of motion, as itis known to be in finding the center of percuffion, which depends on the very fame principles. Tue correction arifing from this change in the procefs amounts only to about 14’, as will eafily appear in the fol- lowing manner: Tue fluxion of the moment of a fphere, from what has been faid already, is 2x3 yx; from the nature of the mirele, x? = 1—y?, as before; therefore x 9 =—yy, x3 x= 3 4 j= ys and 2px3yx=— apxyty—yy, whofe fluent is +4 f, when y = In perce | In a cylinder, y = 1, therefore the fluxion of the moment == 234; whofe fluent is 3 f, whenw =1. Tue moment of a ring revolving round its center is double the momentum of the fame ring revolving round one of its diameters. For let x be the fluxion of the arch, y the ordinate, and x the abfcifla, radius being unity; zy? is the fluxion of the moment of the ring revolving round one of its A ' ; Wee: diameters; but, from the nature of the circle.) s=—, therefore zy? = wy, which is the fluxion of sthe area ABR; therefore when x = 1, that is, when the arch is equal to 1. 4, the meafure of the moment wiil be the area of a quadrant; and the mea- fure of the moment of the entire ring will be equal to the area of the circle, or 2p. Ir the ring revolve round its center, in an immoveable plane, its moment will be equal to the ring multiplied into the {quare of its radius, that is, equalto f Therefore the moment in the former cafe is to that in the latter, as 3 f to f, or as one to two. Hence, from what has been demonftrated, the momentum of a fphere is to the momentum of a cylinder, revolving round their axes with the fame angular velocity, as 4. to 3; the mo- mentum of a cylinder is to the momentum of a ring revolving round its centre, in like manner, as 5 f tom; and the momentum Vou. VII. B of a ® * 5 [re J of a ring revolving round its centre, is to the momentum of the fame ring revolving round one of its diameters, as two to one; therefore compounding thefe ratios, and ex eguo the momentum of a fphere revolving round its axis, is to the momentum of a ring revolving round one of its diameters, as 8f to 15 m, or as 800000 X quantity of matter in the fphere, to 1000000 x the quantity of matter in the ring. Ir therefore g’ 7” 20", viz. the quantity of the preceflion, which according to Newton’s calculation arifes from the aciion of the fun alone, be encreafed in the ratio of 925725 to 800000, it will become 10’ 33”. Bur it is well known, that the true quantity of the preceflion, arifing from the action of the folar force, is nearly double this quantity. Since therefore the correCtion of this 3d Lemma will not account for the great difference between the refult of Newton’s calculation and the truth, we muft look for the caufe of the difference elfewhere. Simpfon is of opinion, that it arifes from this, that the momentum of a very flender ring revolving about one of its diameters, is only the half of what it would be if the revolution were to be performed in a plane, about the centre of the ring ; and therefore, that all conclufions, which do not take this into the account, mufi be two little by juft one half. But itis evident, that this cannot be the true caufe of the difference, becaufe Newton did aCtually confider, that the motion of a ring round one 3 ' | ; ae - foe | of its diameters was lefs than when it revolved round its centre, though he has differed from Simpfon in the ratio which he has af- figned of their motions in thefe two cafes; and when the ratio of their motions is admitted to be as one to two, and the other cor- © rections propofed by Simpfon are alfo made, the total error on thefe accounts is found to be but 1,5”, as has been already» fhewn. Mr. Miner, in his paper on this fubject in the 69th vol. of the Philofophical Tranfactions, agrees with Frifi in thinking, that the error lies in Newton’s affumption, that the -recefflion of the nodes of a rigid annulus and a folitary moon, revolving in the perimeter of the annulus, are equal; whereas in truth, as they affert, (though erroneoufly, as we fhall prefently fhew), the receffion of the latter is but one half of that of the former. Let us therefore examine particularly whether the receflion of the nodes of a rigid annulus be indeed double the receffion of the nodes of a folitary moon, as has been aflerted. Let AE (Fig. 1.) reprefent the rigid annulus, indefinitely flender, projected into its own diameter, P £ its axis; let the line of ‘the nodes be at right angles to SC, the line joining the centres of the fun and earth. From C take the arch C tse and draw LM parallel to DB; let =the gravity of any given quantity of matter, as a cubic inch; °h ="the fpace deftribed-in 1” by a Baal 4 a body * Boare. og body falling freely by the force of gravity; = the periphery - of a circle whofe diameter is unity; alfo let AC = 1; S. angle DCA=s; Cos. DCA =c; arch CL=8; fine of CLay. Then L M = cy, and CM = sy. Tue difturbing force of the fun is equal tof. 1. M (Cor. 17. Prop. 66. Lib. 1. Princip.) and the force of a particle of matter at L to move the annulus about the centre, in the direciuon | PQAD, is CM x f x LM, acting by the power of the lever CM;; that is, the force of this quantity of matter at Lis = esfy?; therefore the fluxion of the force of the matter in a quadrant of . woes ssva9 e annulus is esfy Sx Me ital (sie; iene x is 42 —1y x 1—y™, and therefore the whole fluent is Zesfa—zesfy x i—y".; and when y= 1, the force of the ° . CS) matter in a quadrant of the annulus is = Lt, and the force a of the whole annulus is pcs/= to the fimple force Ee of acting 2 at the diftance /: from the centre, that is, at the diftance of the centre of gyration from the centre of the annulus. This is the force of the fun, to difturb the annulus, when at the greateft diftance from the nodes; call this fimple force Fes. THE Pa ang i aa Tuer quantity of matter in the annulus is 2, and the diftance of the centre of gyration from the centre of the earth is + ; and by the property of that centre, if the whole matter of the an- nulus were collected into that point, any force applied to move it about the centre C, would generate the fame angular velocity, in the fame time, as it wou!d do in the ring itfelf. And fince this force Fes acts at the fame diftance “+ from the centre of the annulus, it is the fame thing as if it were directly applied _ to the body to move it. Now to find the motion generated, fince the {pace defcribed in a given time, is as the force directly, and the matter moved inverfely, therefore g: h:: Bees : pel 2p 2/2 § = the fpace defcribed by the centre of gyration in 1. And 2 p /x (the circumference of the circle whofe radius is the diftance of the centre of gyration from the centre of the annulus): 360°:: i I , nen mee 1 360 Xx ce the angle through which the ring is drawn oa ¥ 4 ' in 1 by the action of the fun, when at the greateft diftance from the nodes. | Bur the force of the fun when at any other diftance from the nodes, as at H, will be lefs; and the mean quantity of the force may thus be inveftigated. Draw the great circle HGP, and making radius = 1, let the arch CH = , fine of CH =y; then in the fpherical triangle CHG, Rad. (1): S.CH (y):: S. angle DCA (s): S. HG = sy. But it has been already proved, that Lists sal ’ that the force of the fin is equal to F x by the produé& of the fine and cofine of his height above the plane of the annulus, therefore the force of the fun at H is equal to F sy X 1—s* y!% But this force acts entirely in the plane PG H4, therefore we muft refolve it into two forces, one acting in the plane POA, which is that we are looking for, the other in the plane PC f, perpendicular to the former; this latter force is deftroyed by an equal and contrary force, when the fun is equidiftant on the other fide of the line of the nodes; but the other force always acting in the fame direQion, is that only by which the | ring is annually affected. The Cos. GH: Cos. angle DCA:: Rad. : Sin. angle H (Cas. 11. Sph. Trig.) and Rad: Sin. angle H:: Sin. CH: Sin. CG (Cas. 2.) ».* Cos. GH (1—s? y*!#): Cos. DCA Cc : (ej Sims CH (7); sin. CG — ras Then, to find the part of the force acting in the plane PQA, Rad. (1.): FsyV1—s* y* * (the whole force):: 8. GC Oma ): Fesy?, the force in the bs Wis direttion PO, And hence to find the mean annual force, we muft find the fum of all the Fcsy* in the circle, or the fluent Md Ms sy2y : of Fesy> z= ts ead 2 Fess—}Fesy /i—y*; and when y = 1, the fluent becomes + F csp, and in the whole circle = Fesg; this divided by the whole circumference 2, the mean force comes out 7F cs, that whofe fluent, found as before, is Per J that is, half the greateft force, when the fun is at the greateft diftance from the nodes. Now to compute the force of the fun to produce the anti- cipation of the nodes of a fingle moon at A, the nodes of the orbit being in quadrature; the force of the fun = fcs; the hfes quantity of matter in the moon is= 1. Then g: b:: feos: —— the fpace defcribed in 1"; and 2 (the circumference of a circle whofe radius is unity, or the diftance of the moon from the earth): 360°:: ones : 360 x oe = the angle defcribed in 1’ by the plane of the orbit of a folitary moon in fyzige. Anp by a-procefs exactly fimilar to that. ufed before in the cafe of a rigid annulus, it may be fhewn, that the mean force of the fun to difturb the moon, conftantly in fyzige, is but half its force when at the greateft diftance from the nedes. Ir follows therefore, from what has been demonftrated, that the greateft force of the fun to move the annulus. in the di- rection PQA is equal to its greateft force to move the plane of the moon’s orbit, the moon being conftantly in fyzige, and that the mean force in both cafes is half the greateft force; confequently the mean force of the fun to move the plane of the annulus in the direGtion PQA ié equal to its mean force to move the plane of a folitary moon in fyzige, in the fame direction: | [ 6 ] direftion. But by Cor 2 Prop. 30. Lib. 3. Principia, in any given pofition of the nodes, the mean horary motion of the nodes of a folitary revolving moon, is juft half the horary motion of the nodes of a moon continually in fyzige. And Mr. Landen, in his memoirs, has fhewn, that when a rigid annulus revolves with two motions, one in its own plane, and the other about one of its diameters. half the whole motive force acting upon the ring is confumed in counteraiing the centrifugal force of the ring, by which it endeavours to revolve round a momentary axis, in confequence of its two motions; and the other half only is efficacious in producing the angular motion of the ring about its diameter; fo that the motion of the nodes of a detached ri.id annulus, being produced by half the mean folar force, is exactly equal to that of the orbit of a folitary moon. For in the cafe of a folitary moon no cen- trifuzal force to produce a revolution round a momentary axis can take place, ‘there being nothing for the body to act upon; but in a rigid ring, its two motions compounded will give the ring a tendency to revolve about an axis neither perpendicular to nor in the plane of the ring, and therefore this axis cannot be permanent; fince each particle of the ring will act by its centrifugal force to imprefs on it a new motion about an axis perpendicular to the former. But if the rigid annulus, fo revolving, be attached to the equator of a fphcre, the cafe will be widely different; for the whole motive force is here em- ployed in giving motion to the annulus and {phere together about * * Poy: | about a diameter of the equator; therefore the part of it which isemployed in giving motion to the ring, bears a very {mall proportion to the whole force, and it is this fmall part only which is countera&ted and rendered inefficient; for the {phere itfelf has no centrifugal force, whereby it endeavours to re- volve round a momentary axis. Hence the motive force being given, viz. the force on the ring, the angular motion generated will be inverfely as the inertia of the matter moved; now the inertia of the annulus is = the matter of the annulus x vz (the diftance of its centre of gyration from the centre of the ring); and the inertia of the fphere and ring together is = the matter in them x 2 ; therefore the angular velocity of the ring muft be diminifhed in the ratio of the inertia of the ring to the inertia of the ring and fphere together, in order to have the angular velocity which now will be produced in the ring, in confeguence of its connection with the fphere, by the coun- teracting force. That is, if a be the angular velocity of the ring and fphere united, the angular velocity which that part of ‘the force which is counteracted could produce in the ring inertia of the ring I Bos Py LL)! The ago part ee be —% aera of the {phere 250 therefore of the whole force only is now efficient in moying the ring round its diameter ; but this part is = the centrifugal force, and therefore it is this part only of the whole folar force which is counteracted. ~ Vou. VII. C Hence @ ae 9 Hence therefore it appears, that Newton rightly fuppofes the preceflion of the nodes of a rigid, detached annulus, and of a folitary moon to be equal; though the principles on which he argues are infufficient, becaufe he did not, as was neceflary, confider the operation of the counteracting centrifugal force. And when he comes to apply this deduction, his conclution is erroneous, becaufe, omitting the confideration of the centri- fugal force as before, he conceived, that the motion of a folitary annulus and of a ring attached to a fphere, were produced by the fame efficient force ; whereas in this latter cafe, the cen- trifugal force of the annulus vanifhes, and therefore the whole force of the fun becomes efficient; that is, the efficient force in the cafe of a ring adhering to the equator of a globe, is double the efficient force in the cafe of a folitary ring; and therefore the quantity of the preceflion, eftimated on this falfe hypothefis, comes out too little by juft one half. : * Brsnop Horsexy, in his commentary on this problem, ob- ferves, that if this affertion, to wit, that the motion of the nodes of a rigid annulus and of a folitary moon are the fame, be true, he cannot fee how the quantity of the preceffion of the equinoxes can be different from that which is affigned by Newton; but he refrains from any abfolute decifion: “ Si hoc “ vere di@tum fit (fays he) f{*. quod par eft ratio nodorum “ annuli lunarum terram ambientis, five lune ille fe mutuo * contingant, five liquefcant, & in annulum continuum for- “56 wlentur; & Cato mentur, five denique annulus ille rigeicat, & inflexibilis “ reddatur, nefcio qui fieri poffit, ut alius fit puntorum equi- “ no¢tialium motus a vi folis oriundus, quam calculi Newtoniani ‘ fuadent. Quem tamen longe alium invenere viri permagni ‘ Eulerus & Simpfonus noftras, quos velim leétor confulas. “ Tpfe nil definio.” Now from what has been faid it clearly appears, how the motion of the nodes of a folitary moon and rigid annulus may be equal, and yet the quantity of the preceflion affigned by Newton erroneous in the ratio of one to two; the efficient motive force of an attached annulus being double the efficient motive force of a ring revolving folitarily, with a compound motion round its centre and one a n . e ¢ of its diameters. * Ir then the corrected quantity of 10’ 33”, be further cor- rected, by augmenting it in the ratio of two to one, the refult will nearly agree with the quantity invefligated by other emi- ‘nent mathematicians; thus Simpfon makes it 21’ 7°, Landen 27" 7", D’Alambert 23° nearly; Euler 22”; Frifi 214°; Milner 21 6’, and Mr. Vince, 21" 6"; fee Phil. Tranf. vol. 77. From this review of the folutions of this problem, it appears that Mr. Landen has the honour of having firft deteG@ted the particular fource of Newton’smiftake, by difcovering that when a rigid annulus revolves with two motions, one in its own plane and the other round oie of its diameters, half the motive force gis -qpates oe acting . 4 5 oe > = ‘4 .? ‘ Pees acting upon the ring is counteracted by the centrifugal force arifing from this compound motion, and half only is efficacious in accelerating the plane of the annulus round its diameter. As Mr. Landen has not exprefsly demonftrated this propofition, I am perfuaded I fhall afford the mathematical reader much gra- tification, by here laying before him the following very elegant demonfiration, communicated to me by the learned Mr. Brinkley, Profeffor of Aftronomy in the Univerfity of Dublin. Prop. If a rigid ring #g NO revolves with two motions (fig. 3.), one in its own plane, and the other about the diameter q FQ; and ifa motive force, acting at the point Q., be fuppofed equivalent to the whole motive force acting upon the ring, then half this force is efficacious in accelerating the motion of the point © (in a direction perpendicular to the plane of the ring) and the other half is confumed in counteracting the cen- trifugal force, arifing from the motion of the particles of the ring about a momentary axis P Tp. In the great circle 7 let a point & (fig. 3.) be taken inde- finitely near to , and in the ring a point 7, fo that 4 and Or may reprefent the angular velocities about the diameter and the centre of the ring. Let d and c reprefent thefe velocities, and y the radius of the ring. Draw rs perpendicular to: the -plane of the ring, and meeting the great circle 4Qs in s; “§ then * Psaor | then will rs reprefent the accelerating force of the point Q, perpendicular to the plane of the ring; but rs: 24:: Or: Rad. (r), therefore rs = = ConsEguentiy, if R = the matter of the ring, a motive force acting upon the point Q = _ = R will be equivalent to the whole efficacious motive force on the ring. Tue momentary axis PT is in a plane perpendicular to the plane of the ring, and which paffes through Qg. * Make PT = the radius of the ring, and draw Pr perpendicular to Qg, and we have Banas. c. OL bk 7 — sai SE pre a Lene Vc* +d* Vc* + d* (in fig. 4.) reprefent the momentary axis, and QEN a quadrant of the ring. From any point E of the ring draw Ev perpen- dicular to P T, and vw perpendicular to QT. The centri- fugal force of E: centrifugal force of N:: Ev: NT, or the Ev ¢* +d? Wir when becaufe the velocity of N = c+ +a. But the centrifugal force of EB = - centrifugal force of N+ == x Ev NT’ : efficacious part of this force in a direction perpendicular to the particle E x : vw : 3 plane of the ring = whole x Eo? and a force a&ting at © equi- valent [ 2 ] “% * aD ae Sata Ev vw Male a Oe ea Soe cy XE X yp X x a 2 = x Ex 2 2 To fac Now if great circles be conceived drawn through P, Q, and P, E; (by Sph. Trig.) cos. PE (vu T) x Rad. (TQ) =cos. PO (Tr) x cos. QE (Tx). -There- fore a motive force at O equivalent to the motive, efficient, cen- } ce +d: lag) are) tor rae Bere trifugal force of E = ae Mee TO: ; therefore the fum of all thefe quantities = the motive force at Q equivalent to the fum of all the efficient centrifugal forces, or the centrifugal force of the ring. But it is eadly fhewn, that the fum of all thefe quantities = —— x5R~x Tr SS aereee = Cig zR (RIT pate ARO) (a Sabine : S&S, . Hence. the.motive. force at 'On Xo 4+a4°x TO! r : Q equivalent to the fum of all the efficacious centrifugal forces, is expreffed by the fame quantity a 2zR, as the force at Q, equivalent to the whole motive, efficacious force on the ring. Q,E. D. . Mr. Srmpson has pointed out the miftakes in the folutions of this problem propofed by M. Silvabelle and Walmefley ; but neither is his own calculation entirely faultlefs; and his conclufion appears to be correct, only becaufe the errors in the premifes com- penfate each other. Thus he fuppofes, that the whole motive force, [ 23 ] force, acting on a detached rigid ring, revolving with a two-fold motion, one round its centre, the other round a diameter, is equal to the efficient force by which the plane of the ring is moved round its diameter; whereas the former is to the latter as two to one; half the whole motive force being counteracted and rendered inefficient by the centrifugal force. 2dly, He fuppofes, that the whole efficient motive force, acting on a detached rigid annulus, revolving in the fame manner as before, is equal td the whole efficient motive force acting on an annulus, attached to and conneéted with a fphere, which is alfo falfe in the ratio of one to two; the centrifugal force in the cafe of an attached annulus vanifhing; and therefore no part of the whole force is rendered ineffectual; and confequently half the motive force in the latter cafe will produce an equal effet as the whole in the former, half of the force in the former cafe not contributing in any degree to the motion of the annulus round its diameter, but being totally employed in counteracting the tendency of. the. ring to revolve round a momentary axis. Mar. Mitner’s and Frifi’s calculations become likewife corre& in the refult, in the fame manner as Simpfon’s, by the mutual countera€tion of equal and contrary errors. ‘Thus they both hold, that the preceffion of a rigid annulus is double that of a folitary moon, whereas they are equal, as we have already de- monftrated, by which the preceffiom would come out twice greater than the truth; but they likewife are of opinion, that the pre- - ceffion [ 24 ] ceffion of an attached and folitary annulus are equal, whereas the | former is double that of the latter; this error therefore counter- balances the former. » Mr. Emerson has given two folutions of this queftion, which are both erroneous, one in his Mifcellanies, the other in his Fluxions. In the former he adopts the fame principles with Newton, in fuppofing the preceflion of a folitary moon, a de- tached rigid annulus, and an attached annulus to be equal. In the latter he determines the dire@tion in which a body would move in confequence of a uniform motion impreffed on it in - one direftion, and a uniformly accelerated motion in another, to be the diagonal of a parallelogram, whofe two fides reprefent the {paces defcribed from quiefcence, in the fame time, by the two forces; which, as Mr. Milner has juftly obferved, produces an error of one half in the conclufion. For let AD be the fpace defcribed by the uniform motion (fig. 5.), while the body would defcribe AB by the accelerated motion; fince the time is indefinitely little, the accelerating force may be confidered as conftant, and therefore the body will in fact defcribe the parabola AGC; and the direGtion of the motion at C will be the tangent EC; but the angle DEC = DAC + ACE = 2DAC nearly, becaufe the tangents AE, CE, are very nearly equal (Ham. Con. Cor. 1. Prop. 3. Lib. 2. and Prop. 3. Lib. 3.);. that is, the true angle of deviation DEC, is very nearly double the angle of [ a5 ] of deviation DAC, as determined by the diagonal of the pa- rallelogram. In this folution Mr. Emerfon fays, “ the earth being an oblate “ fpheroid, the {phere is encompaffed with a folid cruft going “ round the equator in the manner of a ring; now the effect of * the forces of the fun and moon upon this cruft, and the motion “ communicated thereby to the whole body of the earth, is what “ we are to enquire after.” He then calculates the force of the fun upon the annulus, and fuppofes this whole force efficient; he next fuppofes this whole motive force to act at the diftance of the centre of gyration from the centre of the earth, and thence deduces the motion generated in the plane of the equator about one of its diameters. It appears therefore, that he fuppofes the whole motive force of the fun to be efficient on the annulus, fe- parately confidered: and 2dly, that this efficient force is equal to the efficient force on the fame annulus, when conneéted with the earth; which, exclufive of the error dete€ted by Mr. Milner, are the very fame falfe hypothefes with thofe adopted by Simpfon. Bur here a queftion naturally arifes, if the error of Newton’s calculation be as great as is pretended, whence comes it to pafs that the refult of his calculation agrees fo exaétly with phe- nomena; for on fuppofition, that the preceffion arifing from the force of the fun alone is but 9° 7, the preceffion caufed by Vot. VII. D the - [ 26 ] the moon will be 40’ 52” 52", and the whole preceffion, arifing from both caufes conjoined, will be 50’ 0” 127. according to obfervation. : » To this obje@tion a fatisfactory anfwer is fuggefted by Newton himfelf, where he fays, that the preceffion will be diminifhed if the matter of the earth be rarer at the circumference than at the centre. The reafon of which is evident from what has been already demonftrated, for the quantity of matter in the earth _being given, the diftance of the centre of gyration from the centre of the earth will be lefs, the more the matter of the earth is accumulated towards the centre, and therefore the lefs will be the angular motion generated by the fun and moon. + LO fiice LEI "0, 5 ae GENERAL DEMONSTRATIONS of the THEOREMS for the SINES aud COSINES of MULTIPLE CIRCULAR ARCS, and alfo of the THEOREMS for exprefing the POWERS of SINES and COSINES by the SINES and COSINES of MUL- TIPLE ARCS, to which is added a THEOREM by help whereof the fame METHOD may be applied to demonftrate the PRO- PERTIES of MULTIPLE HYPERBOLIC AREAS. By the Rev. J. BRINKLEY, 4.12 ANDREWS’ Profeffor of Aftronomy, and M.R.I. A. "T Heorems by help of which the chords of multiple cir- cular arcs may be found in terms of the chord of the fimple arc were firft given by Vieta, and afterwards in a different manner by Mr. Briggs, which are very fully explained in the Trigonometria Britannica, and their ufes in conftrudting trigo- nometrical tables fhewn. From thefe may readily be deduced theorems for the cofines of multiple arcs in terms of the cofine of the fimple arc, and for the fines in terms of the fine of the fimple arc when the multiplier is an odd number, and _confequently the feries firft given by Sir Ifaac Newton for the fine of a multiple arc when the multiplier is an odd number, the only cafe in which that feries terminates—Afterwards fimilar ; D2 theorems Read May 6, 1797: Bike el theorems for the fine and cofine of multiple arcs, when the mul- tiplier is any whole pofitive number even or odd, were given by ~ feveral authors—But all the writers on this fubje& that I have feen, except Dr. Waring, have deduced the law of the feries from obfervation in a few inftances without a general demonftration of its truth Dr. Waring has (Curv. algebr. Propr. Theor. 26 & Cor.) by help of his admirable theorem for finding the fums of the powers of the roots of an equat. given a general de- monftration of the feries for finding the chord of the fupple- ment of a multiple arc in terms of the chord of the fupple- ment of the fimple arc, and confequently a general demon- {tration of the theorem for the cofine of a multiple arc in terms of the cofine of the fimple arc, and alfo of the fine of a multiple arc when the multiplier is an odd number. But in the cafe where the multiplier is an even number no demontftration, as far as I have feen, has ever been given by any author. Dr. Waring’s method of demonftration cannot be applied to this cafe—The following demonftration extends to every mul- tiplier whether even or odd. ‘The demonftrations for the fine and cofine of the multiple arc in terms of the cofine of the fimple arc, from whence the other theorems are immediately deducible, are of this kind—The probable law is deduced from obfervation in a few inftances and then the general truth of that conjecture is proved. »Dr, Waring’s demonftration, although by a very different procefs, being founded upon the properties of algebraical equations, is alfo of this kind, as it depends upon “= 1S le 2 upon his theorem for the fums of the powers of the roots of an equation, of which he has given the fame kind of demon- {tration—Previous to the demonftrations of thefe theorems I have given a demonftration of the theorems for exprefling the fine and cofine of multiple arcs in terms compounded of the fine and cofine—Thefe theorems alfo have been given by many authors, and the only general demonftrations of them have been deduced from the hyperbola and the confideration of impofflible quantities—However ufeful impoffible quantities may be in difcovering mathematical truths they ought never to be ufed in {trict demonftration, and it muft feem a very circuitous mode to apply the properties of the hyperbola to demonftrate thofe of the circle—Thefe demonftrations are from the properties of the circle and the theorems for combinations. Tue theorems hitherto mentioned are more particularly applicable to the conftruction of trig. tables and the refolution of certain equations—In confequence of the great advances that have been made in phyfical aftronomy fince the time of Sir Ifaac Newton, it has been found neceflary for facilitating the calculation of particular fluents to exprefs the powers of the fine and cofine in terms of the fines and cofines of mul- tiple arcs, and theorems for this purpofe have been given by feveral authors. They have all however either deduced the general law from obfervation without demonftration, or gene- rally demonftrated it by help of impoflible logarithms—The demonttrations fe ger ol demonftrations here given are general, and deduced from the circle by help of the do-irine of combinations. As the hype bola has been fo frequently ufed to demonftrate properties of the circle, I have fubjoined a theorem by which the conneciion of multiple circular areas, and multiple hy- perbolic areas is more fully apparent than by any other that T have met with, and from whence by the doctrine of combina- tions, theorems may be deduced for hyperbolic areas fimilar to thofe of the circle. I. Theorem. Let s and c be the fine and cofine of any arc a, then, radius being unity, and 2 any whole number, ae 7 asics 1. WU—I. 2 i.) Dhetime ofna sce nse ee ee ip Bes Deine 2 m.nm—I n—22 9. The cofine of 2a = ¢ — ——— s+ &c. & I.2 # In each the powers of s increafe by 2, and thofe of c diminith by 2, till the laft becomes 1 oro. In the fine the coefficient of mu—U VU c S =e (to v terms +m. n—l.n—2 - - ken ?—! Neen nnn EEE . I. 2.3 - 2-200 ean is even and—when odd. And inthe cofine the coefficient of s =+ n. n—t1. (to v terms ae. + when —— is even and — when odd. RE: 2 Demontftration ferige Demonftration—Let a, a, a’, a", &c. reprefent any arcs sy $5 5’, 5°, their fines ¢, ¢, ¢, ¢’, their cofines ‘Then by the common theorem for the fine and cofine of the fum of two arcs, The fine 2 of were go se The cofine co—ss The fine S dia t see tsce ts dcomss's fofatadta’= thin ‘ere i The cofine 3 COCO —CSS —COSS &c. &ce The following obfervations may be readily made by confidering the way which in thefe fucceflive values are formed. 1. In both fine and cofine of the fum of x arcs ie +.a' + a &c. the number of factors ss ---cc in any term is equal to # and that the fines s, 5, s’, &c. and alfo the cofines c,c,c, &c. are concerned exattly alike in the whole quantity. , 2. In the fine of the fum of » arcs (a + a’ + &c.) the greateft number of cofines c,¢,¢c, &c. together in any term = »—1I. This number diminifhes by 2, and confequently the number of s,s’, &c. increafes by 2. 3. In [ “3@2.] 3. In the cofine of the fum of # arcs the greateft number of c,c,c", &c. in any term = 2, the next lefs number z—2, &c. and confequently the number of s, s, &c. increafes by 2. 4. WitH refpeét to the figns of the different produ€ts—In the fine of x arcs a+a +a’ + &c) when 1, 5 0r4p+1 (f being any number, s,s, s” &c. are united together, the fign is + other- wife—. In the cofine of 2 arcs when 2, 6, 10 or 2 £ ( p being odd) s,5,5° are united together the fign will be — otherwife +. 5. In mo term can the fine and cofine of the fame arc occur. 6. In any termss'5" --- cc'c’ --- whether of the fine or cofine if m be the number of the cofines and confequently m—x the number of the fines: then, becaufe each of the quantities s,s’ &c. and alfo c,c &c. are concerned exactly alike in the fine of the fum of # arcs (2 + a + a’ + &c.), and alfo in the cofine of the fum of # arcs (a+ a + a” + &c.) and likewife be- caufe the fine and cofine of the fame arc cannot occur in the fame term, it follows that the number of terms ss‘ s” (m terms) - - - ccc’ » -- (m—wn terms) = the number of combinations of z things taken m together = 2. n—1. n—2 - - - u—m—tI Hot en Ae RMSE oS ™m FROM E33 J From thefe obfervations it immediately follows, if aka, na—l 2. n.—T' n—2 a’, &c. are all equal, that the tie nme re oS ee i. 2ae n—2 2 c s t+ &c. and ness 2 n—tI cs + &c. and that the cofine of za =c—x. 4 alfo that the general terms are as ftated in the theorem. Q,E.D. | : nm—1I7 n—3 n—-2 “TI. TureoreM. 1. The cofine of 2a=2 ¢—n. 2 ¢€ + se ais ae 8c. to be continued by fucceflively diminifhing the Ip 2 : ; amu index of ¢ by 2 till it becomes 1 or 0, and affixing toc the coeff. fm tae HERE Lie ES Ah u n—u—1 2.Nn—u t+ l.n—u+2.- - to > terms +2 ; . of which the fign is 1 ea Binet tas. “, 4+ when 2 is even; and — when odd. Aa—I ni—tit n—3 a ats 2. The fine of na=2 c— m—2.2 ¢ + &e. :WI—c? continued by diminifhing the index of c by 2 till it becomes n—u 1 or o, and affixing to c the coefficient nan AUPE. N—U+2- - (ane terms) pte A A of whiclithe fign I. a. - - u—t 2 is + when «+ 1 isodd and — when even, 2 > Vou. VIL E DemonsTR. E iat al Demonstr. By fubftituting in the values of the fine and cofine of xa found by the laft theorem, for x fucceflively 2, 3, 4, &c. and exterminating s it may be conjectured that the ge- neral terms of the fine and cofine will be as here ftated. That this conjecture is true’ appears in the following manner: N—I—u NU Let Be be a term in the cofine of z—1 a, and Cc Be ey en Bie a pert: 9 V71—e*, and De /i—c? terms in the fine of z—1 a: and that the latter terms will be of this form appears from the former theorem. Applying the common theorem for the fine and cofine of n—uU the fum of two arcs, it readily appears that the coeff. ofc _ in the cofine of na = B—C+D. Now fuppofing the theorem generally true and fubftituting in the general terms for x, n—1 and for w fubft. wu, w—1 and un + 1 fucceflively, the refult is —_- —_— a“ n—u—z N—I+ 2—u. n—u + 1-- - - to ri terms B= it2 > : si a“ Te. te 12 eu uay HFT? Spberegs . 2 RE eet u Tee Fo a ae Sb — —1I terms n—y N—u + 1. N—uU + 2 to 4 ms —C=t2 x : i ce peat Se a ——1 terms se agi to > D2 Eas 7 aE yl ae u 2—1—2 N—U—I. m—u =~ =-+to — ferms D= bd 2 x : “ a 2s) s) Ook tame pO! terms, 2 (ay ey SA reer (es eros Od : n—u—t RL eco { u—u Aes oe — = uw B—C+D 4+2 xX 2.4———I rx 1 — 2 I I Z.= n—u—t wie | [2X —u—l. a—u 4 Be 12>) 0.8 2 9 terms 2 — ieee a = ——I 2 Gains: _ Suge Seer u n—u—1 2.N—u-+ 1.N—u+t 2 = eat EO x, terms. + 2 x u hin Se _- . - — 3 2 n —u—tI Let alfo Ge s be a term in the fine of n—1. a, and let — n ee sis A He be a term in the cofine of ~—1 a, and it readily appears m—tU that G+ H=coeff. of the terme ‘5 in the fine of xa. Now fuppofing the general term of the fine truly expreffed, * E 2 G= es | ee a u~—tI rasta asa i aT Oy —— ‘terns = ie een Pa SO dee ts f e 7 u—t Wes Fe a ees Ce TORE LET Os 2. 3 = bea iby Wot ote hee So ee to is. terms H= ae 2 DO ee Ea u—t1 Bate Dew Ge SE ene 7 terms eee eS u—I nu 3 .2n—u—I. n—u+I - to terms 2x . u—I Tet 20S Paes ; terms n—u n—u-+1.n—u + 2 - - to —-— terms nate oie aad Paneer See mmmeersetiret se pe Net Ee a u—t = * vd ies) ‘ i] to —-— terms: 2 Hence it appears that if the general terms are rightly exprefled for the fine and cofine of paras they are alfo rightly exprefled for the fine and cofine of 7a, confequently if they are true in the inferior values of they are true in the fuperior, but they are true in the inferior *.. &c. &c. > Ill. Cor. If the feries be arranged in a contrary order ; \4- +4 2 y = | nN ie) >» 1. WHEN 7 is even the cofine of 2a = " I. 2 1.2.3. + &c. and the general term is + ¢ where v is always even. When ~ is of the form 24, (p being any odd number the fign will be + or — according as = is odd or even and when z is of the form 49, (g being any. number) it 4 Vv will be + or — according as Z 18 even or odd, wv ae 2. WHEN 2 is odd, the cofine ofma=+tney™ pie TB, . Ty ee 2 2 2 v-+- I 2.1—I, Nm——3, =e Se terms » and the general term is + be Tt. 2 3 - let 4 where v is always odd. When ~ is of the form 49+ 1 the fign' BEATYs . will be + or — according as — is odd or even, when of the v+I form 4+ 3 it will be + or — according as is even or odd.. e Each feries is to be continued till the coefficient becomes = o. Dem. The general term of the cofine of. #4.. bx aeeyl aol >... Spee e ee ee u m—u-1 2. 2—u + 1. w——uw+2 - = terms s—u =+2 2p ee RE NS SSAA Ok RS il} ¢ u epee 8 - - 2 terms or fubftituting for u, z—v, the coeff. becomes +e ae fs na+u—4, n+v0—2 Pi 2 sag ie. 2 pyghny pst ey 2 n.N—V—2. n—vV—4. - N+ V—4. N+ V—2 VET Tosh say Rade 555 oe wee, vw—tI us + 2 2 2, , Tey mae a 3 - Vv + 2. 12—U—2. N—V—4 - - n21+U—4. t+ U—2 we Tay Be oe. ae - v 1. WHEN 7 is even and «vw even it is of this form ae 2. 2—V—2 - - u—2. 2.n+2 - - n+Uv—2 I. 2. Bi i ae 22 AW ale & Vv = tn.nu—2. u—4 - - to > terms he 3 - - v i i u—v , Tue fign is + or — according as > or —z is even or odd. Poe is [sow] . Ir 2 be of the form 24, (f being odd) the fign is + or — 2 p—v . : is according as is even or odd .*. as 7 3 oddoreven. If z —v be of the form 4 / then it is + or — as a2 dita is even or odd U and °.: as > 1s even or odd. 2. WueEN zis odd and + w odd the gen. coeff. becomes of 2. A—VU—2 - - a—iI. am+I = = 2+ U—2 1 Onl 68a aS ee ea Mn) Deed - - - U CMe ee pa hee v+I . TE IE ere, Fb > = to - terms. a ie paedeads 2 - = = v . . “ . Tue fign is + or — according as 7 Of 7 is even or odd. co . Ir w be of the form 4+1 it is + or —as ‘eer or- sper even or odd or .*. a8 °FT is odd or even. If x 2 be of the form 44 + 3, itis + or —as aD spBBSE oie aI Wi acai hai 2, v+i or .", a8 ——>— is even or. odd. Whence &c, &e, IV. THEeorsM: [40st | IV. THeorem. 1. When wis any even number. The fine n—i n—I pet n—3 of 2a Shes Ry | 2. (mols oa eee: mas to be con- tinued by diminifhing the index of s by 2 till it becomes unity. The upper figns take place when zis of the form 29 (p being odd) and the lower when it is of the form 4p (p being any number). SP etn Coens u—tI N—u--I. 2—uU +2 - to aaa terms Tue general term is + eo 2 1G, ae Cg Pe nu n—u et Spoil } x2 5 X ¥1—s*: + when — — is odd and of the form 2p (p being odd.) “wel, ~ _ when ee Bi iseven w+t, + when —>— 1s even Sand # of the form 4 (pbeing any number) — when“ is odd 2. WHEN # is any odd number, the fine of 2a = + m—l 1 I—3Z i—Z2 2 s 422 5s -+&c. to be continued by diminifhing the index of s by 2 till it becomes unity. The upper figns take place when x is of the form 4+ 1, and the under when of the form 46 + 3. THe oar J pe w f.m—y+l. nw—u+2 - to Be terms Tue general term is + ————________ oe I. ° 3 = ci Ya 2 3 2 A—u—TI n—uw, a> 5 4+ when = is even p and w of the form 4f+ !. —when = is odd | when > 5 “ + when = is odd | : > and # of the form 4 + 3. — when Bl is even | DrmMon. The general term of the fine of 7 x Q—a = (II) “u—I M—U41. 2—U-2 - - OLS cs Si AR = eee? Qa" u—I I, 2. 3 ~ =F ‘mem ~ x s, Q—a, where Q.is a quad. 1. Let abe of the form 2%, p being odd. The fine of 2p tiple of the circumference) fine 2 Q—2pa=s, 2pa .*. when n is of _ the form <4, p being odd the general term of the fine of na= Vou. VII. FE As MW—u+1. A#—uU+2 = rok terms »—u \n—xz eat ga.” ee u—lI I. 2. - ros as 3 2 PT tel a + when is odd and — when even. Let 2 be of the form 49, £ being any number. Tue fine of 4p X O—a = (becaufe 4 Q is a multiple of the circumference) = fine of —4pa=—s, 4pa.. when a is of the form 4/ the general term of the fine of na=+ a u—T BK ET ep RUT gee geal eee te X2 5,@ Xc¢sa—when — u—ti Tiare - - z ett is odd and + when even. 2. WHEN is odd. Tue general term of the cofine of # x Q—a = (Il) “ 4. An I. n—u eo =) tog POS os Sy re =e u I. 2 is - sal 3 eo n be of the form 4p + 1. _ THE cofine of 4AptixQ@ =a =e 4p04+Q-4ptia= — a o———_——__ ‘ faes Lh eA Ds wad SI es, Q—4f+14= fine 46+ 14.. when a is of the form 4f+1. the ad: | the gen. term of the fine of xa= a“ We N2—ut I. n-—u-+ 2 - - to 7a terms + sles 5 I. 2 ” = + = = No 3 z “oan ae u : 2 5, 2 + when 2 is even and — when odd. : Let z be of the form 4/ + 3. Tue cofine of 4pt+3 x O—a = cs of 30—46+34 = (becaufe adding or fubtracting + the circumference changes the fign of the cofine) = — cs of Op +3 a=—s. of 4p +3 a ' . WHEN n is of the form 4p + 3 the general term of the fine of Site eS eee ul 2. A—uU+ 1. N—ML2 - to — terms g—y—, a 5 2 2 sav naa 3 1 MESO ps Rar — when * is even and + when odd. Whence the truth of the theorem will eafily appear. ¥V. Cor. If the feries be arranged in a contrary order. 2 2 nt 3 Tur fine of nA = ng — 5-6 + &c. when » is any odd F 2 number [44 ] number; and the fine of na=nys — “" "sy 4 8c. x /1—s* when 2 is any even number. In the former cafe the general term is P =} 2 2 Uv + I =< =a = = Weaesenl 1.2—1I. 2—3 to » terms v ae s me Gen - = = to v terms v+tr : v being always odd, + when —z 1s odd and — when even. In the latter cafe the general term is + 2 2 2 2 Vv -- T to ——— terms % 2, Uv . 5 xVI—s, + when > 1s I.2.3 - - tov terms odd and — when even. Turs Cor. may be deduced from the theorem in the fame manner as the Cor. Art. III. was deduced from its theorem. Theorems for the Powers of the Sines and Cofines. VI. Turorem. If c be the cofine of the arc a and rad. unity then z being any whole pofitive number. 7 Ro = n—I n+. e= 3\ x: esna+tun.cs n—2 a + &e. cont. to | terms when z is odd and when a is even to 4 2+1 taking only + the lat Pd [i ae J * 2-1, 2—— - - to m terms Batt, CEPEn ioe meneneral TEN 75. ee Se Be Bate hee hea gg) cS A-—-2 Mm a. Dem. Let a, a’, a’, &c. reprefent any arcs c, ¢, ce’, &c. their cofines THEN by trig. cs, a X 2 ¢5, a= cs, a+ a +05, a—a and in like manner cs,aX2¢5,a'x2c5, a"=cs, a4-a'+¢5,a—da' y 2¢5,a” = es,a4a+a'+ ¢s,a+a—a'405,4a—a ba" + cs,a—a—a’, bce. &e.. and it is evident that to multiply by twice the cofine of any arc it is only neceflary to encreafe and diminith each of the former quantities a + a’ + Sc. a—a', &c. by that arc, and take the fum of the cofines of the arcs fo encreafed and diminithed : therefore becaufe in the produé of the cofines of a, 2a’, 2a", &c. all the arcs a’, a’, &c. muft be involved exa@lly alike, it follows that rad | 2 % C5, aXcs, a'Xcs,a' xX &c. = fam of the cofines of all the arcs formed by adding to 2 each combination of the »—1 arches a’, a’, &c. taken pofitively or negatively. Hence by the theorems | for combinations, there will be 1. term the cofine of a+ a+ a” + &c. i—t1 a [ 46 ] z—1 terms the cofine (fum #—1 arcs = 1 arc) (B) “Tl: #2 terms the cofine (fum 7—2 arcs —fum 2 arcs) (C) I. 2. ——___ (JS a -= i=—™m a ee. ee terms the cofine (f{um of wz—m arcs — fum m arcs) (H) ee Se eas j —_____—_ ._____ terms the cofine (fum m arcs — fum I. 2 - m—i n—m arcs) (H’) nz—1 terms the cofine (fum 2 arcs — fum 7—z2 arcs) (C’) 1. term the cofine (1 arc (2) — Sum z—1 arcs) B’. Now if the arcs be all taken equal, all the Bs are equal to each other, all the C, &c. &c. and alfo B= —B,C=—C &e. &e. and confequently cs, B=cs, BY, cs, C=es,C’, &c. &e. “nT. = n—m =I - 2—m—1 I. 2.- I. 2e 3 - m—tI = Aj l= a i ————— CS, 1—2 ma. WHENCE Bovey oth « n at Ye Sept —— WHENCE c= i) * cs matncsm—2zat+n.a—i csn—4art &e. I. 2 “ n+ 5 i is h continued to —_— terms when is odd: but when 7 1s even there 2 . . ara rrr n will be a middle term n—1I- w—2 - - > n X es Pema Be) Bers omy ate 4 a —_— nm #.7—1 - - to 2 terms aE X es 0a *.* in this cafe 2.1.2: 3-°- Fn n n—tI1 n a eT ee ees 1 ec=2'Xesna+n cs m—2a + &. to > terms Aer elie n Bias | terms. 2 n TE abs cna ee ad 2 VII. ToeoremM. 1. When-# is any odd number, and s the “n n—I fine of any arc. a, rad. being unity,s =2) x +s5,na} 2. 5,2—24 bee: n—4 4} &c, continued to ——— terms &c. the upper Bez 2 figns taking place when z is any odd number of the form 4p + 1, and the lower when of the form 49 + 3. THE ‘eer! 2. m—t1 - (to’mterms Mier ie = m f : th is + comes — > THe general mt. term is S; N—2m a + when # is even i and z of the form 4+ 1. — when m is odd her is odd + when m 13 0 i and # of the form 4/ + 3. — when # is even 2. WHEN z is any even number. n t—T Crue. n s=i) x +esnatn.cs n—2at &c. (to = terms) + n—=I SORRY : 2 Xm mts 2 = 22 terMS The upper figns take place Dial Xie Da hE mi = ee when z is of the form 4, and the lower when of the form 2, p being any odd number. The m4. term is +n.n—I - (mterms -——— SS ng i Se I. 2.3 - (mterms) + when m is odd i and # of the form 2, p being odd. — when m is even +- when 7 is even ken Pineda ¢ and n of the form 4. —wWw Ahn a tse n—~] at oe Dem. Let Q=a quads. then (VI) cs. O—a'= 4) xesm. O—a d . %.N—1. - to m terms 4 &c. and the general m+, term is I. 2 3 - mterms hole cS A—2m, Oma. Tegtit ex aay | 1. 1ft. Wuen 2 isof the form 49 +1, fubft. for #, 4p +1 cs n—2 m. Q—a=cs, 4p—2mQ + Q—n—2ma = (becaufe adding or fubtraéting the circumference makes no alteration in the value of the cofine and adding or fubtra¢cting 4 the circum- ference changes the fign of the cofine) + cs O—7—2 ma= + 5, n—2ma + when m is even and — when odd. 1. 2. Wuen 7 is of the form 4p+ 3, fubft. for 7, 4p + 3, cs, n—2 m. O—a=cs,4p/+3—2mO—n—2ma—t 5,n—2 m4, + when m is odd-and — when even. , g. 1. WHEN 7 is even of the form 2), » being odd, fubft. for — 2p, cs 2—2m. O—a=cs 2p—2m O—n—2ma = tcsn—2ma, + when m is odd and — when even. Qe Qn Wuen nm is of the form, 4p; fubftituting for 7, 4, cs N—2 m. O-—a=ces 4 p—2 m, O—1n—2 ma=tesn—2zm, a + when 7 is even and — when odd. Wuence fubftituting in the general term for the cs, Q—a, the s, a and for cs, n—2 ma, the values above found, the truth of the theorem is evident. Vou. VII. G THEOREM Bago Propertres of the Equilateral Hyperbola. VII. Turorem. Let a, a, a’ reprefent abfciffas meafured from the centre on the axis of an equilateral hyperbola, and 0, 0, 0 correfponding ordinates: let alfo the hyperbolic area contained by the femi axis (= unity), diftance from the centre to the extremity of the arc, and the arc, the abfcifla of which is a” and ordinate o”, be equal to the fum of the areas contained in the fame manner ‘by the femi axis, dift. and arcs the abfciffas and ordinates of which are a, a and o, o: then will a” = aa’+ od and o’ =ad+da'o. Dem. ° ‘Let the area ACV (fee fig.) = ECV + BCY, let the double. ordinates FEe, 45GB, aHA be produced to meet the affymptote C w'x'y/ NY X mz W 4, and let fall the perp’. aw’, dx’, oy, VN, EY, BX, AW. Becaufe ACV = EVC + BCV and becaufe (by prop. hyperb.) CVN = ECY = BCX = ACW VNEY + VNBX=VNAW or VNEY = BAWX: andit has been proved by many writers on conics that when thefe areas are equal CN CY CX: CW or WN Be: BX: AW Whence it follows that - CV: Em:: Ba: Ap ca ‘ u a“ Or lI: a—oO-;.- 4—6- @—0 E2ot 4 J in like manner it may be fhewn that CV:: em:: bn: ap or li ato::dto:a'+o hence a’—0" = ad—aco—do-+o0 and a’ +0’ =ad+ad4a0+ 00 and «.:a@’=aa +ooando'=ao+ do. Q,E.D. From the fimilitude between thefe theorems and thofe for the fine and cofine of the fum of two circular arcs, it is unneceflary | to point out how every thing may be deduced for multiple hy- perbolic areas in the fame manner as was done for multiple circular arcs. ee: a REMARKS on the VELOCITY with which FLUIDS sfue from APERTURES 77 the VESSELS which contain them. By the Rev. MATTHEW YOUNG, D.D.S.F.T.C.D. 8 MRL A. > Ww HEN water iflues from a fmall aperture in the bottom or fide of a veffel, which is kept conftantly full, it has been fuppofed, that the force accelerating the loweft plate of water, of indefinitely little altitude, immediately over the orifice, is the weight of the incumbent water only; and therefore, that after the motion of the plate has once commenced, the preflure of the incumbent column will be diminifhed, and of confe- quence, the force accelerating the plate, during its defcent through its own altitude, will not be conftant. But, in fact, it is mot the preflure of the incumbent water, which accelerates the loweft plate; for every plate of water immediately incumbent over the hole, abflracting from all lateral preflure, begins to be accelerated equally at the fame moment; and therefore the incumbent column, exclufive of any lateral preffure, could produce no increafe of velocity, in proportion to its increafed height. The force which really accelerates Read Jan. zoth, 1798. TM oa accelerates the iffuing plate, is the preflure of the ambient water, which furrounds the cylinder immediately over the aperture ; and this lateral preffure being communicated to the upper furface of the plate, muft be as much encreafed by the velocity of the fuperior defcending plate, as it is diminifhed by that of the inferior iffuing plate, fo as to remain conftantly of the fame magnitude. * Ow this principle it can be eafily demonftrated, that. the velocity with which water fpouts from an aperture in the bottom or fide of a veffel, is equal to that which a heavy body would acquire in falling through the height of the fluid above the orifice. Turs demonftration, however, as Mr. Atwood obferves, is true only on hypothefis that the water fuffers no refiftance, but iffues in a cylindrical or p ifmatic form correfponding to the hole. But, in faét, the velocity of the water according to theory will be dimi- nifhed by the friction of the particles againft the edges of the orifice ; from their mutual attraétion, by which the iffuing particles are retarded by thofe which are {till in the veffel, and have not acquired the velocity of thofe which precede them; but prin- cipally from the obliquity of their motions. For, as Chev. Du Euat obferves, when water iffues from an orifice, the particles will flow from all fides, towards the orifice, with bee 4 with an accelerated motion, and in all directions. If the orifice be horizontal, that filament of particles, which anfwers to the centre of the hole, will defcend in a vertical line, and will fuffer no other refiftance than that of the friction caufed by the excefs of its velocity above that of the collateral filaments, or by the retardation which arifes from the attraction fubfifting between them. The other filaments, after they have defcended vertically for fome time, are compelled to turn from their vertical courfe, and to approach the orifice in different curves; and when they arrive at it, their direCtions become more or lefs horizontal, according as they pafs nearer to or farther from the edge of the orifice. The motion therefore is decompofed according to two directions, the one horizontal, which is deftroyed by the equal and contrary refiftance of the filaments which. are diametrically oppofite; the other vertical, in proportion to which the quan- tity of water difcharged is to be eftimated. Hence we fee, that the vertical velocity of the filaments decreafes from the centre of the orifice to its circumference ; and that the total difcharge is lefs, than if all the filaments had iffued vertically, in the fame manner with that which anfwers to the centre of the aperture. It alfo follows, that the filaments which are nearer to the centre, moving fafter than thofe which are nearer to the edges, the vein of the fluid, after it has iffued from the orifice, will form a cone whole bafe is the orifice; that is to fay, that its diameter will diminifh, at leaft, to a certain diftance, becaufe the exterior filaments are gradually drawn on, in confequence of their j mutual |g So mutual attraétion, by the interior filaments whofe velocity is greater; whence there follows a diminution in the diameter of the vein. ; Tsis manner of accounting for the contra@tion of the vein feems more reafonable than that which is given by Newton; as there appears to be no adequate caufe for the acceleration of the water, after it has been difcharged from the orifice. Tue diminution of the mean velocity of the water, caufed folely by the obliquity of the motions of the iffuing particles, exclufive of any other impediment, may be thus determined: Let mx (fig. 1.) be the diameter of the aperture in the veffel ABDC filled with water: in whatever dire@ion the water iffues, its velocity in that direion will, in all cafes, be the fame, becaufe the preflure of fluids is the fame in all direiions ; thus, whether a fluid fpouts perpendicularly upwards or downwards, horizontally or obliquely, the fpace through which it is projected, in a given time, is the fame. Now to determine this direGion, fince the horizontal and vertical preflures are equal, the iffuing particles will aflume the intermediate dire@tion, which will therefore form an angle of 45°. with the plane of the orifice: its vertical velocity therefore will be lefs than its dire@ or total velocity in the proportion of the diagonal of a fquare to its fide, or sy | or as 7 to s nearly; but the particles of the central filament iffue with the full velocity, due to the entire height of the water; therefore the velocity of the central particles will be to the mean velocity, as 7 to the mean between 7 and 5, or as 7 to 6. This is the diminution, as has been faid, which takes place in confe- quence folely of the obliquity of the motions with which the particles iffue from the orifice: if the other caufes of retardation be taken into the account, we may conclude, that the velocity fhould be diminifhed perhaps in the ratio of 8 or even g to 6; which accords very well with experiments. Thus Polenus makes the ratio of the diameters of the contracted vein and aperture, which is the fame with that of the mean and greateft velocity, to be as 53 to 63; Bernouilli 5 to 7; Chev. Du Buat 6 to 9. When the orifice is infinitely little, the cylinder of iffuing water becomes a fingle filament, which is therefore difcharged without any obliquity, and there will be no diminution of velocity, except fuch as arifes from fri€tion and the tenacity of the par- ticles. If the aperture be encreafed fo as to become equal to the bafe of the veffel, the column of water will then defcend like a falling body, and therefore the velocity will be the fame as before ; but it will not acquire this velocity until the uppermoft plate of water has been difcharged. At the beginning of the motion, the firft or loweft plate will flow out with a velocity indefinitely little ; the next plate with a greater velocity; and fo on, until the upper plate fhall have defcended to the orifice which will then iffue with the greateft velocity. But if the Vou. VII. H veflel be a5 82a] veflel be fuppofed to be kept conftantly full, the velocity of the effluent water will encreafe fo as at length to become equal to that which a heavy body would acquire in falling from an infinite height. Since the middle filament of particles is difcharged with the full velocity due to the entire altitude of the fluid above the orifice, experiments made on the diftance or height to which fluids fpout, will be found to agree very well with theory, but it by no means follows, that all the filaments fhould be difcharged with the fame velocity: the quantity of the ffuid therefore difcharged in a given time, may be lefs than that which would be difcharged, if all the filaments were difcharged with the velocity due to the entire altitude; becaufe this quantity de- pends on the mean velocity of all the filaments. Hence there- fore it cannot be inferred from thefe experiments, compared with thofe which relate to the height or diftance to which the fluid fpouts, that the velocity of the water in the orifice is lefs than that which is due to the entire altitude; and that it is accelerated immediately after it gets out of it: becaufe the diftance to which the fluid fpouts, depends on the central filament only, but the quantity difcharged on the mean velocity of the whole. To bring this queftion to the teft of experiment, if all the particles were equally accelerated at their difcharge from the orifice, and immediately after they leave it, they ought all to be projected * ga . 2% : projected horizontally to the very fame diftance upon an_hori- zontal plane, but ‘on experiment I found, that.when the fluid fpouted through an orifice of ,o8 of an inch diameter, and was kept conftantly at the fame height, the greateft and leaft diftances at which it ftruck the horizontal plane were nearly 15 and 12 inches; but thefe diftances are proportional to the velocities with " which they are difcharged. It follows therefore, that all the particles are not projected with the fame velocity. It is to be obferved, that the particles which are difcharged with the greateft and leaft velocities are few in comparifon of thofe which are difcharged with intermediate velocities, for while the entire fhower extended from 15 to 12 inches on the horizontal plane, the denfer part was found to occupy only the fpace between 14% and 124 inches; fo that the limits of the velocities of the parts of the denfer fhower were as 7 and 6,26; but the limits of the whole were 15 and 12, or as 7 and 5,6; and the limits by theory are as 7 and 5. But we may perceive, that when the fluid fpouts horizontally, the particles which iffue from the ‘upper part of the aperture, and which therefore ought to move with the leaft velocity, muft encounter thofe below them moving with a greater velocity, which will encreafe the diftance to which they are projected on an horizontal ac Likewife, the particles which iffue from the loweft part of the orifice, and which ought to move with a lefs velocity, than that with which thofe in the axis move, in the ratio of 5 to 7, will have their velocity en- creafed by their being at a greater depth. The limit therefore of the ratio of the diftances to which the particles are projected Vou. VII. *H 2 on 60 ‘ ee es on an horizontal plane, muft be Tae than that which refults from the theory of water iffuing through an Piorizonul aperture. But it is obvious that the greater depth of the lower particles, when the Orifice is vertical, cannot account for the entire difference of diftance to which the particles are projected; for the depth of the orifice being 8,55 inches, and the diameter of the orifice ,08 of an inch, the velocities on account of the difference of depth- would be only as /8,55 to /8,63, or as 14,6 to 15 nearly. Perhaps it might be faid, that this difference of diftance was caufed, not by the different velocities, but by the different direGtions in which the particles are difcharged; fo that thofe which are projected in the axis of the vein, will ftrike the ho- rizontal plane at a greater diftance than thofe which are projected from the edges of the orifice with the fame velocity, but in a different direGtion, But this cannot be the caufe; for when the aperture is horizontal, the particles which iflue from the oppofite fides m,n of the orifice (fig. 2.) meeting each other, deftroy their convergence, and afterwards proceed in the direction of the axis of the vein, and therefore the vein will continue nearly of the fame diameter: whereas, if the particles croffed each other, with the fame velocity, in different directions, they would defcribei inter- feéting parabolas 1s, m/, and the diameter of the vein would con- tinually encreafe. In order to determine whether this were the cafe, I caufed the fluid to iffue through an aperture in the bottom of the veflel, and at the diftance of 12 inches I found the diameter of the vein a little encreafed, when the velocity of the efflux was con- . Giderable ; [ 61 } fiderable ; but not fenfibly augmented, when the velocity was much diminifhed. Since the dilatation of the vein in this cafe de- pends-on the velocity with which the water iflues from the aperture, it is to be inferred, that it is caufed by the refiftance of the air; which producing a retardation of the preceding particles, thofe which follow impinge againft them, and the thicknefs of the vein is encreafed ; for the fame reafon as when the jette.is made perpendicularly upwards, a broad head is formed in confequence of the retardation of the uppermoft particles. Now fince it ap- pears, that the dilatation of the vein which arifes either from the different directions. of the particles, or the refiftance which they undergo from the air, or both together, cannot: account for the difference of diftance to which the particles are projected on an horizontal plane, we muft conclude that this difference is caufed by the different velocities with which they efcape from the- orifice: Wuaen a'tube murs (fig. 3) is inferted into the veffel ABCD; it is found, that the velocity. is increafed nearly in the fub-duplicate- ratio of the length of the pipe, when the tubes are fhort; and that it approaches nearer to that fub-duplicate ratio, according as the length of ‘the pipe is increafed. account for this increafe: of velocity has appeared.a matter of fome difficulty, fince the water cannot iffue at rs with a greater velocity than it enters at mn, and it does not appear how the velocity at m7 can be en- creafed by inferting a. tube beneath it. In order. to -explain the caude - ts 4] caufe of this effect, we are to confider, that the whole force with which the plate mx is preffed down, is the weight of a column of water equal to emn/f, together with the weight of a column of air of the fame bafe, reaching to the top of the atmofphere; and the whole force with which it is preffed up, is the weight of an equal column of air, diminifhed by the weight of a column of water equal to mars. therefore the actual force with which the plate mz is preffed down, is, the weight of a column of water equal to efrs; the velocity therefore with which the plate mz will iffue through the orifice mn, will be the fame as through the orifice 7s in the veffel AdcD; that is, equal to the velocity which a heavy body would acquire in falling through the altitude er; and all the plates of water in the tube mars will defcend with the fame velocity; for they cannot defcend fafter, becaufe other- _ wife there would be a vacuum left in the tube, which is pre- vented by the upward preffure of the atmofphere. And the velocity of the effluent water will be the fame, whatever be the preffure of the atmofphere, provided the weight of a column of air of the fame bafe with vs, and whofe height is equal to that of the atmofphere, be either greater than or equal to the weight of the pillar of water murs. This might be proved experi- mentally by a veffel of water with a pipe inferted in the bottom, placed under an exhaufted receiver. But as the operation of ex- hauftion is obf{truéted more by the evaporation of water than of mercury, it will be better to ufe mercury in thefe experiments. Now if D be the defe&t of the gage from the ftandard altitude, it will pes will meafure the preffure of the air on the furface of the mercury in the veffel; let A be the altitude of the mercury in the veffel above the upper orifice of the pipe, and P the length of the pipe; _ then the whole force preffing downwards the plate of mercury which is immediately in the upper orifice of the pipe, will be = D+ A; andthe whole force preffing the fame plate upwards. will be D—P; and the difference between thefe forces will be the abfolute force preffing the fame plate of mercury downwards; while D is greater than P, this abfolute force will confequently be: equal to A+ P; when D=P, D—P vanifhes, and the force: prefling the plate downwards is = D+ A=P-+ A; hence therefore- no variation in the time of the efflux will be perceived, while the altitude of the mercury in the gage is equal to or lefs than the difference between the length of the pipe and the ftandard altitude. When D is lefs than P, the force upwards is alfo nothing; and therefore, as before, the whole force preffing the plate down-. wards is = D+ A; and A being given, it decreafes according as D decreafes; and when D vanifhes, that is, when the receiver is: abfolutely exhaufted, the force becomes equal to A, and the time of the efflux will be the fame, as if the pipe had not been in- ferted in the bottom of the veffel. To try the truth of thefe things by experiment, I inferted a tube 7,8 inches long in a cy- lindrical veffel, and clofing the orifice of the pipe, I filled the ‘veffel with mercury to the height of 6 inches; then placing the apparatus under the'receiver of an air-pump, when the barometer was. at 30 inches, and the gage at 28,5, the time of the efflux was jer was 26 feconds; when the experiment was repeated precifely in the fame manner, but in the open air, the time of the efflux was only 19 feconds. Now as the gage ftood at 28,5, the defe@ D was 30—28,¢ = 1,5, and the preffure on the plate of mercury was = 64 13 = 72; in'the-open air the preflure was= 64 7,8= 13,8; therefore the ratio of the velocity of the efflux in both cafes, which is the ‘fame with the reciprocal ratio of the times, was “7+ to 13,8, or as 2,73 to 3:7; but 2,73 is to 3,7 as 19 to 26 very nearly. No difference was obferved in the times of the efflux, when in the open air and exhaufted receiver, unlefs the gage ftood higher than 223 inches; that is, unlefs the height of the mercury in the gage was greater than the difference between the length of the pipe and the ftandard altitude. In another ex- periment, when the gage ftood at 27,9, the height of the barometer was 29,9; the defect therefore was — 2, and the preffure = 8. But V8 = 2,828, and V13,8 = 3,7 but 2,828:3,7::19:24, and by experiment the time of the efflux appeared to be 23 feconds. When the efflux is made in vacuo, it is obvious to obferve, that the pipe is not filled during the efflux, as it is while the difcharge is made in the open air. Since the column of water in the pipe mars adds to the preffure which the plate mz fuftains, by diminifhing the upward preffure of the air through the pipe, it appears that it produces this increafe of preffure in the plate mz alone, without producing any ~ bea any lateral preffure in the water which is on a level with ma for it is manifeft, that if an aperture were made in mB or 2C, the velocity of the water iffuing through it would not be affected by the infertion of the pipe; and confequently that the plate mx, which is immediately in the orifice of the pipe, is the only one, on the fame level, whofe tendency downwards is increafed by the infertion of the pipe. Hence, the particles of water at the edge of the aperture, having their perpendicular preffure encreafed by the weight of the column mars, without any increafe of their, lateral preffure, they will iffue through the orifice mm more per- . pendicularly; the fides alfo of the tube will obftru€t the con- - verging motion of the particles, and confequently, on both thefe - accounts, the quantity of water difcharged through a pipe thus inferted, will exceed that difcharged through a fimple orifice, in a greater ratio than the fub-duplicate of the height of the water. . And according as the length of the pipe encreafes, the ratio of o. the quantity of water actually difcharged by experiment, to that which fhould be difcharged according to theory, will increafe; becaufe the ratio of the perpendicular to the horizontal preffure _increafes, in the ratio of the fum of the depth of the veffel and - length of the pipe, to the depth of the veffel. It follows therefore, that experiments ‘made in this manner, will approach nearer to coincidence with theory, than when made with a fimple orifice; except either when the tube is fo long as that the friction of the fluid againft the fides of the tube fhall produce 2 fenfible effet, or Vow. VII. Vee when... a ©) Loreen] it is fo fhort, as not to be fufficient to give the particles a vertical ~ dire@ion. All which agrees very well with the experiments made by the ingenious Mr. Vince, of which he has given us an account in the Phil. Tranf. for the year 1795. Thus he tells us, that having inferted a tube, a quarter of an inch in length, into a cylindrical veffel 12 inches deep, he found that the velocity did not fenfibly differ from that through the orifice; the caufe of which he difcovered to be this, that the ftream did not fill the pipe, but that the fluid was conttaéted, as when it flowed through the fimple orifice. When the pipe was half an inch long, inferted into a veffel of the fame depth as before, the velocity of the water from the pipe and from the orifice, which ought by theory to have been as ¥12,5 to W12, or as 49 to 48, was by experiment found to be nearly in the proportion of 4 to 3. Now if the ratio of 49 to 48 be increafed in the ratio of 7 to 6, (becaufe this is the ratio of the diminution of the velocity on account of the con- traction of the vein, and this contradtion either nearly or entirely vanifhes in a pipe,) we fhall have the ratio of 3,57 to 3. When the pipe was an inch long, the velocity from the pipe and from the orifice, which, according to theory, ought to have been as /13 to /12, or as 26 to 25, appeared by experiment, very nearly in the ratio of 4 to 3; now if the ratio of 26 to 25 be encreafed in the ratio of 7 to 6, we fhall have the ratio of,3,64 to 3. When he made ufe of longer pipes, the velocity of the effluent water by experiment approached nearer to that which ought to have been igh £O7 « 1] been difcharged according to theory; fo that in long pipes, the difference between theory and experiment, he fays,: was not greater than what might be expeéted from the frition of the pipes, and other caufes which may be fuppofed to retard the velocity. When he inferted a pipe of the fame diameter with the aperture, which terminated in a truncated cone fixed “in the orifice, (fig. 4.) he expected, that the quantity of water difcharged in a given time would have been diminifhed, becaufe the water, iffuing through the orifice mz, would have room to form the vena contraéta in the enlarging cone; but he found, that the fame quantity of water ‘was difcharged, as if the pipe had con- tinued throughout of the fame diameter with the orifice. The reafon of this is manifeft from what has been faid, for the preffure ef the air will not fuffer the truncated cone to remain partly empty, as it would be if the vena contracta were formed; it will therefore continue full, and confequently the water will pafs through it in the fame manner as if the water in the cone, furrounding the pipe maz, were congealed. Mr. Vince likewife inferted into the bottom of the veffel a perpendicular pipe, in form of a truncated cone, the narrower part being fixed in the orifice ; by which he found the efflux to be encreafed more than if he had inferted a cylindrical pipe ef the fame length, whofe diameter was equal to that of the narroweft part of the conical pipe. This effect may be explained on the fame L232 principle [ 68 ] principle by which we accounted for the augmentation of the: diameter of a vertical vein of water, through a fimple orifice, when the velocity of the efflux is confiderable. For when a per- -pendicular pipe is inferted, the velocity of the difcharge being - confiderably encreafed, the refiftance from the air will be fo like- wife; and thus the diameter of the vein has a tendency to en- large itfelf; now in the widening cone, the pipe admits of this augmentation, at the fame time that it encreafes the velocity; but the cylindrical pipe, though it equally encreafes the velocity, yet it does not permit the vein to enlarge itfelf, and by thus con- fining it, the efflux is obftru@ed, and the quantity difcharged in a given time is diminifhed. Accordingly, under the receiver of an air-pump, even in a moderate degree of exhauftion, there is no difference perceived between the velocities with which a fluid is difcharged through a conical or cylindrical pipe. Gea Bijoncuequns sq s=-25) re eae ’ re . + — E89 7] A new METHOD of refoluing CUBIC EQUATIONS. ~ By THOMAS MEREDITH, 4. B. Trinity College, Dublin. ‘Tue roots of a cubic equation of this form, #3 + 3c.%2+3¢2.% Read June + c}—a=o which differs from a power only in its laft term, aiiatiesh can be found, by tranfpofing, a, and extracting the cubic root on each fide, provided, @, is not an impoflible binomial. PrositemM. ‘To reduce any cubic equation to this form, x? +3¢.%*+3c*%.x+c*—a=—o, that is, to reduce it to an equa- tion, in which, the fquare of the co-efficient of the fecond term is triple the co-efficient of the third, Ir the roots of a cubic equation, x* + px* + 9x+r=o, are en- creafed or diminifhed by any quantity, p* and 39, will be en- ereafed or diminifhed by an equal quantity, if multiplied, will be multiplied by an equal quantity, therefore their equality or inequality, not affected by thofe transformations. i aot “3 +px*+gx+r=0 = P4307 4307483 x=y+a pea py +2pay+pa fo A Pp =94a°?+6apt+p and 39.=94*+64p+ 39 therefore p* and 39. encreafed by the fame quantity, viz. 9a* + 6 ap ape +ge+-r=0 Ci es a J? 1 pay? PG a yt a? r= 0 p =p’? @ and 39=392° therefore both multiplied by the fame: quantity, Viz. a. Hence it appears that the problem cannot be effedted by thofe transformations. Bur the equation, x* +px* +9x+r=o by transforming the roots into their reciprocals, and freeing the firft term from a coeffi- cient becomes, x° + 9x* +prx+r* =o therefore if in the propofed equation g* = 3pr, then by transforming the roots into their re- ciprocals, and freeing the firft term from a co-efficient the equation will be reduced to the required form. Any cubic equation being propofed, there is a quantity, by which if the roots are encreafed (or diminifhed) g* will become equal Ligrs | equal 3 fr, the value of this quantity may be inveftigated by folving a quadratic equation. Tuus let the equation be, Rae 1) ae oa Gt ls a lm P+ 307 + 3c +e aI +e py” +2 pey + pe* POR ge Fa fees +69 36 +2pe+g= get + 12 pe3 ~e+4pge+9? +4¢ pai ee ead ae AS aed Oe, 3xei tpe*4+9e4+7%X 3e+p=ger++12pe ek er sir | +3P +39 + 69 +99 +9r ge*+12pe° -¢+4pgetg’® = get +12 pe* te e+ 3pr +4p° TSP, 1 BPP 25 eke RS a Age a é =0 ma fe OT SPE } Ler it be required to find the roots of this equation, e+6x07+3x+2=0. Subftituting in the formula 26 U8 18a 1.9 10 .e eae em OF 5:° Foe TL e avs wa a) ae | 8 GH aeho a6 ae BS 89 «Tames =) 7 6x — 697 + 12746 sel ag a a Oe ae 2 I J =m I> GIVI AEH 42S 0 % a 12 I2v}418v°+9v+1=0 2e+i8z 4 108z+144=0 Extra@ the cubic root on each fide f 23418 2* 4+ 1082+ 216=42 2, 6= 772 Z=— 64249 —642%9 12 12 De te = ex SS 12 —6+2V, " 556 hala ae fubftituting then for 2 /9 its 3 values, 2(/9, —1+V—3 XV/Q, ~ —I—/—3 x./9 the roots of the propofed equation will be 12 12 12 nig a ee —64+2y —6 —1 + ¥—3. vue : —6 I /—3-/9 : Burt the roots of the given equation may be found after one transformation, for the roots of the final equation are expreffed in the co-efficients of the firft transformed equation, and the root ~ be pr J root of the firft transformed is its abfolute quantity divided by the root of the final equation. Ler the equation when its roots are encreafed by e, be A ae) SY +py +qay+rao Te ruitgui'tput+i=o B+tgzetrpset+ri=o *% Nt 54/8. Be 0 oa Zoe 27 I= Va ‘ae 3 Vv 99 “ WEN the value of e, by which the roots are to be encreafed (or diminifhed) is impoffible the coefficients of the transformed 3 equation will be impoffible binomials “4 = oe — yr’ an impoflible binome (unlefs in particular cafes the coeff. of the impoffible part vanithes) hence it appears that a, and ¢, will be poffible or im- poffible in the fame cafes. mV OL.) VIL. K ; Ir * Since the equation 23 + 92*>-+rpz--7* may be thus. expreffed zi + 3. Le + aida eR ey 3 » 27 27 WBiy eee Ir the roots of a cubic equation are encreafed by — = the fecond term will vanifh, if by —t =s vee + the third term will vanifh, therefore if p* = 3 7 the fecond a third terms may be exterminated together, therefore the equation will have two impoflible roots; hence it appears that the equation of the required form has two impoffible roots,* confequently the value of e, by which the roots are to be encreafed will be impoffible when all the roots of the propofed equation are real -.- 4, whofe cubic root muft be computed, will be an impoffible binomial, unlefs in the particular cafes where the coefficient of the impoflible part vanifhes. Ir remains to be proved that when the propofed equation has but one poffible root, the value of e, by which the roots are to be encreafed (or diminifhed) will be poflible and confequently a, Lert the roots be —m+y—%, —m—V/—N, iS p=a2m+b, g=m+n+2bm, r=bm*>+bn fae aia Rea i wee tut =O Tey =P GER a * That the equation of the required form has two impofhible roots, appears alfo * from this, that two of the cubic roots of a, are impoffible. ——" ey cera m. 6 + 2m.-e+m* —2bm —4bm* —2bm} —3n +2mn +4? m* b? +24*m —2bmn —8bua +2nm*- —3 3b +2 Call the coefficients 2, 7,3, oy _ /y¥ Ba 28+ 4G" ~ 2 . - therefore it is to be proved that & — @0is affirmative, and confe- _ quently the fquare root poffible. inde =m —abm? +mnth?ma4br = oe m°—4bm5 +20. m'—12bn. m 41852 2. m2 —S8bn. m+16b3 2? - +667 —463 + n? — 83x i + 54 BO= m—2bm+hm—2bmn + 2nm—3h nt xm —2bm—3n+F= - mS — 46m + 6b%.m4 — 463. m3 — 5n2.m+4b0.m+10b? n? —n + o. +4632 — 3n! einai Y—Gi= 32m —12bn.m* +18 8 2. mm —12bn.m+ 6b + 6n —125'2 +373 : . +35'n K 2 but [ 76 ] _ but this remainder may be refolved into thofe parts. m—4b6m +68 m—4bm+bx3n= m—b* 3n m—2bnm+Px6ne= m—b.6n FS Be c= 33 #, is affirmative, and m—d° and m—bd*alfo affirmative therefore the remainder affirmative. Lert the roots be — m —m—6 m. e+ im. o@me+ ni . m —2bm —2hm —2bm =o + 3 Bath a epee e+2met+m=o- € =—m + Vm —m therefore when two roots of the propofed equation are equal, the value e, by which the roots are to: be encreafed will be one of the equal roots, therefore the two laft terms of the transfarmed equation will vanith, therefore reducible to a fimple equation, which will give the remaining root C.F 0S tes Oo re 6 49- +IrgZ +256 — dee? Le =ove+azet4o=ove=—2+yV 4-4 — 48 —108 —252 — wey 6 + 127-8 es eae, E 9 — 289 + 28 16y— 32 > Ae es Ses ae ee ceo ISH L eR 3, — 2, —2 ¢ ae ( a ay ; 4 nae hd ~ oo - " ‘ aes ’ : ‘ ‘, =s ie + ' ‘ Ser - ‘ ‘ ) x +. sr « 1 se 4 ‘ s = 5 s Pn PA ee, - 4 ey RA SY st ; . j “ # 7 1 ’ a . \ ow oe t , ‘ ‘ ite: #8 ‘ 2 Ae iN ; "oR oa ot Te “4 : : \° ogee = via : 2 - ’ 2 m ‘ 7 > is Fi , i é x » P| E98 3] On the FORCE of TESTIMONY 7 effablifhing FACTS con- traryto ANALOGY. By the Rev. MATTHEW YOUNG, D.D. - See aG: Dy SMR LA. Tl n’eft pas fi glorieux a l’efprit de Geometrie de regner dans la Phyfique que dans les chofes de Morale, fi cafuelles, fi compliquées, fi changeantes. Plus une matiere lui eft oppofé et rebelle, plus il a d’honneur a la dompter. / FONTENELLE. ARISTOTLE obferves, that the chief charaGteriftic of quantity Read, Feb.3d 7 clay : : 4 8. is, that it is. that by which any thing may be denominated egua/ me F. _ and unequal. Pred. p. 34. Ed. Sylb, Every thing therefore is faid to admit of quantity, which is capable of more and lefs. Hence quan- tities are reduced to two claffes, thofe which confift of parts, and thofe which are eftimated by degrees: accurately fpeaking, the former alone are quantities, the latter are fo only metaphorically. « Propter fimilitudinem dicuntur quantitates, quantitas perfec- “ tionis, quantitas virtutis; intenfionis, valoris, & fimilium.. ‘In « his enim eft fimilitudo quadam quantitatis, que in eo pofita © eft, quod ficut quantitas molis dicit extenfionem quandam par- ee “ tium Fe Soi) |] “ tium extra partes, ita & dictee quantitates fuo modo quandam “ extenfionem partium habent. Smigl. p. 294.” This latter fpecies of quantity is therefore called “ Quvantitas Intenfa & Virtutis,” Aldrich p. 43; for the effential perfeGtions and virtues of things are compofed of different degrees, in the fame manner as quan- tity, properly fo called, is compofed of parts. Burgefd. p. 21. Quantities which confifts of parts are alone capable of meafure, and therefore of mathematical comparifon ; while the others, though they admit of more and lefs, yet not being meafurable, cannot be mathematically compared. Thus different areas, which confift of parts, are meafurable , but pleafure and pain, heat and cold, proba- bility and improbability, virtue and vice, which are eftimated by de- grees, are not meafurable. Crakanthorp therefore defcribes quantity, by faying, that “ it is an abfolute accident, by which things are “« meafured primarily and per fe,” p. 81. Now to make quantities which confift of degrees, and therefore are not meafurable, the fubje@ of mathematical comparifon, an arbitrary meafure is affigned, by referring them to fome meafurable quantity to which they are related. Thus, in the graduation of the thermometer, an arbitrary meafure is eftablithed for heat and cold, for the degrees of heat are referred to the expanfion of the fluid contained in the thermometer, which is meafurable, and to which heat is related. In the fame manner, probability has no meafure in itfelf; but an arbitrary meafure is affigned to it, by referring it to the ratio of the number of chances by which the event may happen or fail; and ° [ 8 ] and thus it becomes the fubje&t of mathematical calculation, in the fame manner as the degrees of heat. Tue ratio of thofe quantities which confift of parts, cannot always be accurately afligned; neverthelefs, fince the quantities are finite, they muft have fome finite, determinate ratio to each other. Thus the area of a circle to the circumfcribed fquare ‘cannot be accurately exhibited: in thefe cafes we can, in general, proceed by continual approximations, and affign limits within which the true ratio muft fubfift. ‘Ir there be two things, one of which is greater or lefs than the other, they are quantities of the fame fpecies: thus when a cannon ball is faid to be greater than an orange, the abftra@ magnitudes of both are quantities of the fame fpecies. Tr two things ‘be of the fame {pecies, and one of them’can ‘be reprefented by an exponent of a given kind, the other is in its nature capable of being expreffed by an exponent of the fame kind. ‘Thus if the area of a fquare be reprefented by agiven right line, the area of the infcribed circle is capable of being reprefented by another right line, though no mathematician has yet been able to fhew what that line is, by any geometrical conftru@ion. Ir the velocity of a ray of light incident on a piece of cryftal be expreffed by a given number, there is a number which will - alfo exprefs its velocity within the cryftal. eVoL. VII. L Tus [paar Tue adfive, efficient caufes of events are thus enumerated in the Ethics of Ariftotle, “ the feveral caufes appear to be nature, ne- “ ceffity, and chance, and befides thefe, mind or intelle&t, and ‘“‘ whatever operates by or through man.” L. 3.c. 3. Chance therefore is an aétive, efficient caufe; but it is alfo am accidental caufe, “ ad caufam per accidens revocatur fortuna et cafus,” Burgefd. Chance therefore is an efficient, accidental caufe of an event. Tue probability of an event, according to De Moivre and Simpfon, is greater or lefs acgording to the number of chances by which it may happen, compared with the whole number of chances by which it may either happen or fail. As, fuppofing it were required to exprefs the probability of throwing either an ace or duce at the firft throw with a fingle die; then there being in all 6 different chances or ways that the die may fall, and only 2 of them for the ace or duce to come upward, the probability of the happening of one of thefe will be 3 or +, Wuererore if we conftitute a fraGtion, whereof the numerator fhall be the number of chances whereby an event may happen, and the denominator the number of chances whereby it may either happen or fail, that fraCtion will be a proper exponent of the pro- bability of happening. For bes 3 For the fame reafon, the probability of its failing will be equal to the number of chances for its failing, divided by the fum of the number of chances of happening and failing together. Tue probability therefore either of the happening or failing of an event is always expreffed by a proper fraction. Ir the number of chances of happening be = 0, that is, if the event be impoffible, the numerator, and therefore the fraction will be = 0; © therefore denotes impofiibility., Ir the number of chances of failing be = 0, that is, if the event be certain, the numerator will be equal to the denominator, and the fra@ion = 1 ; unity therefore expreffes certainty. ProzgapiLiTy therefore extends, as Mr. Locke obferves, from certainty to impoffibility. — _ Wuen the chances for the happening of an event are equal to ‘the chances of its failing, the fraction, exprefling the probability, is = % which is the mean between impoifibility and certainty. One ‘event tlerefore is faid to be more probable than another when its probability is expreffed by a greater fraction; though, in the common acceptation of the word, ‘that only is faid to be pro- bable, whofe probability exceeds half certainty ; for if the proba- wl 2 bility se anal bility be equal to half certainty, it is called doubtful, and if the pro- bability be lefs than half certainty it is faid to be improbable. Since the chances for happening or failing are equal to the | whole number of chances, the probabilities of the happening and ' failing of the event are together = 1, that is, equal to certainty. Tuererore the probability of happening is equal to the diffe- rence between. certainty and the probability of failing; and the probabity of failing, equal.to the difference between certait and the probability of happening. : From what has been faid it follows, that the probability that a witnefs tells truth, in a given inftance, will ‘be expreffed by a fra@ion whofe numerator is the number of chances for his telling . truth, and the denominator the fum.of the numbér of chances. for his ee truth, and for his telling ‘falfhoad together. : “Iw like manner, the probability that an argument is true, is to be efiimated by the ratio of the number of chances for its truth’ to the number of chances for its truth and falfhood together. . ~ Sip: Satts My Iv is true, that in neither of thefe latter eafes” can we, in general, determine the aCtual number of chances 3. _neverthelefs i in all cafes where a perfon perceives the probability ‘of an“event, he muft at the fame time perceive, that there muft be fome finite, Be determinate is them, as the Ariftotelians- did, but to have a fixed. exprefflion, [iss | determinate ratio between the chances: for its happening. and- failing, though he cannot affign: that ratio; for if there were no finite ratio, either the numbes of chances for its happening muft be infinitely greater, or infinitely lefs than the chances -for its failing; in the former cafe, the event would appear certain, in the. latter impoffible, therefore probable in neither. Ir may perhaps be objected, that. if we cannot determine the - a€tual number of chances, all confideration. of the manner. of exprefling mathematically, the probability. of events is nugatory. But it is by no means fo; becaufe though we cannot determine the exact degree of credit, which we ought to give to each witnefs, yet we can determine according to what law our belief ought to’ 5 vary in the cafe of coneurring witnefles, each of. equal credibility. ' Things that are quite unknown, fays Hartley, have often fixed relations to one another, and fometimes relations to things known ; and as,.1n Algebra, it is impoffible to exprefs the relation of the unknown. quantity. to other. quantities known-or unknown, ’till it has a fymbol affigned to it, of the fame kind with thofe that * denote the others ; fo in-philofophy, we muft give names to un- known quantities, qualities, cautes, &c not in order to. reft in under orbit tostreafare up all that can be known of the unknown caufe, bc. #in- the imagination, and » ‘memory; or in writing, for peters ‘gabe Vol? I. p. 348. Pes) We can alfo from thefe principles fhew why after a certain number of witneffes have attefted a fa@, any farther evidence is fuperfluous. Tuese principles likewife, as Dr. Waring obferves, may be applied to the inveftigation of the probability of the truth of the decifion by any number of voters, and many other cafes; the probability of each voter voting truly being fuppofed given. But, as he alfo obferves, it is impoflible to determine the knowledge, integrity, and various influences which actuate each perfon, and confequently to determine the probability of their voting truly. » Bur though we cannot determine the a€tual probability, yet fince the voters are to be fuppofed of equal integrity, knowledge, _ écc. we can determine the relative probabilities of the truth of the decifions by different majorities; and on thefe principles Mons. Condorcet has enquired into the laws according to which the majorities, which decide queftions in deliberative affemblies, ought to be regulated. Tuus fuppofe the enaGting of a new law were propofed to a deliberative affembly, fuch a majority fhould be required as would give a very great probability of the juftice of their decition; for it is much better that no law fhould be enaéted than a bad one. A majority of more than one fingle voice feems alfo requifite in fome queflions of a civil nature, as for inftance in long continued - poffefiion ae [ey] poffeffion ; for though length of poffeflion fhould not fuperfede right, yet confiderable regard fhould be paid to it, not only for the fake of the public tranquillity, but likewife becaufe in the progrefs of time, there, in many cafes, arifes a greater difficulty of producing the original titles of property. So that perhaps it would be wife to increafe the majority, requifite to decide the queftion, according to the duration of the poffeffion. On the other hand, all “queftions which require immediate determination, fhould be decided even by the leaft poffible majority. Ir follows therefore, that although we cannot actually affign the fradion. which exprefles the credibility of a given witnefs, yet our reafonings on teftimony will be rendered more clear, determinate, and extenfive by this notation. And accordingly Dr. Waring, after he lays down the principles for determining the probabilities of events, obferves that they may be applied to human teftimony. See his Effay on the Principles of Human Knowledge, § 17. Tuart probability may juftly be expreffed by a fraction, certainty being denoted by unity, and impoflibility by a cypher, will likewife appear from the following confiderations : Tr upon the happening of an event, fays De Moivre, I be en- ' titled to a fum of money, my expedtation of obtaining that fum has a determinate value before the happening of the event. Ip ae a ‘Ir a perfon therefore tells me, that an event has happened, by which Iam to receive a fum of money, my expectation of re- ceiving that fum has a determinate value, before I certainly know whether that event has actually happened or not. ‘In all cafes, the expectation of obtaining any fum is eftimated by multiplying the value of the fum expected by the fraction which reprefents the probability of obtaining it. Thus if my probability of obtaining L100 be 2, my expeciation will be = 2x f100= f£6o. ‘THEREFORE it neceflarily follows, that the probability of ob- taining the fum is equal to the value of the expectation, divided by the value of the thing expected. And fince the expeétation is neceffarily determinate, fo likewile is the probability. Now my expectation, derived from the report of the witnefs, muft be "either equal to, greater, or lefs than the expectation derived from an equal chance; the probability will therefore be either equal to, greater, or lefs than an equal chance; therefore the probability in the former cafe is homogeneous with the proba- bility in the latter; but the latter is capable of being expreffed fractionally, therefore fo alfo is the former. Suprose a perfon of good chara¢ter tells me, that an event has happened by which I am to receive £100; there will hence arife an expectation jn my mind, which muft be of fome de- terminate [te See J terminate value: for there is a fum lefs than £100, for which I would fell my chance, otherwife I muft confider the report of the witnefs as abfolutely certain; alfo, that there is a fum for which I would not fell my chance, is likewife evident, for if not, I muft have no reliance whatfoever on the witnefs. We can therefore affign limits, within which the meafure of my expeGation fubfifts; and therefore there muft be fome inter- mediate, determinate fum, which is the meafure of my expecta- i i ¥ I | ane Ae. o) tion. Let this expectation be = -- x £100; then > Xx = n I 2 100) exprefles the probability that the witnefs tells truth; or rather is the meafure of my belief in his veracity. Tuis expectation is to be refolved, as Hume and Waring ob- ferve, into the conftitution of our nature; the Supreme Being having impreffed on our minds a faculty for the fource of all our knowledge refpecting exiftence, namely, a neceflary or impulfive belief of the future from the paft, viz. that what has, for the time paft of our lives, been joined together or conftantly fuc-. ceeded each other, will for the future be joined together, or be found in the fame order to fucceed each other. So that having obferved, that in certain circumftances men tell truth, there arifes, by the conftitution of our nature, or as fome hold, by affociation, an expectation, that, in like circumftances, other men will likewife tell truth. Vot. VII. M Bur [ go ] Bur the expectation, in the fame circumftances of an event, will be different according to the conftitution of the expectant ; for, according to his antecedent experience, knowledge, pre- judices, and paffions, the arguments for or againft the proba- bility of the event will appear more or lefs numerous, more or lefs cogent; fo that in given circumftances of an expe and let the probability that a given witnefs tells truth be expreffed by the i . fraction are , 6 being lefs than a; take fuch a power 6” of 4 as that it fhall suib a, and let # be the number of witneffes, then will the probability of the veracity of the concurrent witneffes be n b expreffed by the fraction Pap which is greater than the fraGtion > ;3 becaufe unity, the given difference of the numerators and denominators bears a lefs proportion to the greater quantity 4”, and therefore - the numerator and denominator of the fraCtion [ tos J therefore the quantities 6" and 4°41 are more nearly equal than a ae Ir is manifeft, that where the credibility of each witnefs is very great, a very few witneffes will be fufficient to overcome the pro- bability derived from the nature of the fa@. Thus fuppofe the 6560 ea ; and let us fuppofe that each witnefs tells truth only nine times for once that he tells falfhood ; that is, let the latter probability = probability of the truth of his report be equal only to ~ ; then four fuch concurring witneffes will be fufficient to sae wetted / AFTER a certain number of concurring witneffes have given their teftimony in confirmation of the truth of a fad, any farther encreafe of their number is fuperfluous; becaufe the difference between unity and the fraction exprefling the probability, which is ‘ the refult of their concurrent teftimony, is indefinitely little; and all that an indefinite encreafe of the number of witneffes could do, would be to diminith that indefinitely little defeat. Yet that probability, in cafes of teftimony, admits of an un- limited encreafe, is evident; becaufe the limit of probability is cer- tainty, but the denominator of the fraétion, which expreffes pro- hf always exceeds the numerator by unity; therefore the Vou. VII. O fraction pureoo, “] fraction can never be equal tu unity; that is, no finite number of concurrent reporters can produce abfolute certainty. By this unlimited encreafe is to be underftood the aétual, not {enfible probability , for the indefinitely little defect from certainty is capable of mathematical computation, as well as the greateft quantity, though it be imperceptible by the human mind. We are therefore juftified in concluding, that the evidence of human teftimony effec- tually attains its maximum, becaufe it arrives at fuch a degree, as that any further increafe of it is imperceptible. And the like takes place int extenfion; fuppofe a yard to be encreafed by the hundred thoufandth part of an inch, and by half that quantity, and by the 1, and +, &c. ad infinitum; the increment of this line would be imperceptible, and yet the line would never attain its maximum. Ir the chances for the truth and falfhood of the report of each of any concurrent witneffes be equal, no number whatever of fuch witneffes can render an event probable, by their teftimony. Becaufe the number of chances of their coincidence in falfhood encreafes in the fame proportion with the number of chances for their telling truth. Let their number = #, fince the probability that each witnefs tells truth is = 4, the meafure of the probability of the ; y n concurrent witneffes will be = — 2n Ir it be improbable that each witnefs tells truth, that is, if the — number of chances that each tells falfhood, be to the number — of aes | of chances of his telling truth, in any ratio greater than the ratio of equality, the greater the number of concurrent witneffes, the lefs will be the probability of the truth of their report; becaufe the greater will be the number of their combinations in falfe report in proportion to the number of their coincidences in truth. Thus if there be three witneffes, each of whofe credibility is meafured by +, that is, if there be one chance only for the veracity, and four chances for the falfhood of each, then will the improbability of the truth of their report be meafured by 7. Tuts conclufion, as Mons. Condorcet obferves, ieads us to a very important remark, which fhews how unfit numerous popular affemblies are for deliberation; for fince in fuch affemblies, when we confider the ignorance and prejudices of the voters, we muft eftimate the probability that each will vote right at lefs than an ~ even chance, it follows, that the more numerous the affembly, the greater will be the probability that their decifions will be falfe. And hence we perceive, what political evils muft follow from the deter- minations of an ignorant democracy. But in a well informed and impartial affembly, the more numerous the voters, the greater will be the probability of the re@titude of their decifions. Hence, by the way we may remark, that Dr. Halley’s mode of computing the probability of the report of concurrent witneffes is erroneous. According to him, the calculation is to be made in O 2 the fe: x68" «| ‘ f ¢ a 5 following manner ; if the firft witnefs gives a, of certainty, atc ; ; be aie f and there is, wanting of it mae the fecond attefter will add 4 of that —; and confequently leave wanting only atc aoe : Q@-¢€ 2 2 ‘_.. And in like manner, the third attefter of that ——— a+c a+c- Z : a cz F c3 adds his PEG of that a and leaves wanting only Pro &c. Hence, he obferves, it follows, that if a fingle witnefs thould be only fe far credible as to give me the half of full certainty; a fecond of the fame credibility, joined with the firft, would give me ith, a third jths, &c. which appears to be falfe; for we have fhewn above, that no number of fuch reporters could produce an affurance greater than that of an even chance, for the truth or falfhood of the fact. Tue fallacy of his argument lies in this, that he fuppofes all the individual concurrent witneffes to produce unequal degrees of affurance, which is evidently a falfe pofition; fince they are all of equal credibility and equally concurrent, and therefore contri- bute equally in producing our affurance. Dr. Warine, whofe folution is effentially the fame with Halley’s, fays, if there be two different arguments (or witneffes) entirely independent of each other, in fupport of a fact, whofe probabilities let L ae9 'l let be 2 and 2; then will the probability in fupport of the : a a : fa&, refulting from both arguments (or witneffes) be 1— Cae for if the probabilities in fupport of it are re- a, . fpectively P and £, then will the refpeGtive probabilities of its a a failing be fas Big = ae and r— 4 — Pons and confequently a - @ the probability of failing from both will be <2. —, whence the probability of the fad ae fil both ‘will be r— age a—q a a In this argument there is one ftep, which appears inadmiffible ; it is affumed, that if the probability of ine from both, or rather of both failing, be = “x Grd. then 14 “XE x fea —f= = the pro- _ bability of happening es both, which abs not raya tone" true; becaufe pee ae? is equal to the probability of both a a happening, together with the probability of one happening and the other failing. Thus if there be an even chance for both, paw od = 7 = 4; then + = the probability that both will fail; alfo + = the probability that both will happen, and 2 = the probability that one will happen and the other fail; therefore i—7=%+2=the fum of the probabilities that both will happen, and Pigaso: | and that one will happen and the other fail. This mode of calcu- lation adopted by Dr. Waring, however it may hold in joint an- nuities, where the defired end is equally anfwered, whether one or all of the lives attain the propofed period, will not equally apply to the conjoint probability of arguments, or concurring witnefles, where the evidence fails either when the arguments are all falfe, or are oppofed to each other. Ir the witneffes that atteft a fact, or the voters that decide on a queftion, contradict each other, and it be required to determine what is the refulting probability of the truth of the fa@ or of the decifion upon the whole, we are to proceed thus: firft compute the odds that the affirmative witneffes are right, or the ratio of the number of chances of their being right to the number of chances of their being in errror; proceed in the fame manner with the negative witneffes; then the product of the number of chances -that the affirmative witneffes are right, into the number of chances that the negative witneffes are miftaken, will be the number of chances for the truth of the fat; and the produ& of the number of chances that the affirmative witneffes are miftaken, into the number of chances that the negative witneffes are right, will be the number of chances for the falfhood of the fa&t; and confe- quently the probability of the truth of the fa@ refulting upon the whole, will be equal to the former produ divided by the fum of the two produ€ts. For example, let there be feven voters, of which let four be affirmative and three negative; and let the chance pliawec] chance that each votes tightly be the ratio of 4 to 4, then the ratio of a‘ to 6% will be the odds that the affirmative voters are right ; and the ratio of a to 4° will be the odds that the negative voters are right; and the ratio of a+ 4° to 44 a’, or a to 4, will be odds refulting that the affirmative voters are right. Ir there were eight voters, the loweft majority muft be five and three; and the odds that the affirmative voters were right would be a5 tod’; and the odds that the negative voters were right would be a* to 6* ; and the refulting odds that the queftion was juftly decided would be a‘ 4: to a3 35 or a* to b?. In general therefore it appears, that the odds for the truth of the decifion, will be that power of the odds that each perfon votes juftly, whofe index is the difference between the number of affir- mative and negative voters. Anp hence we may correét the error of thofe who imagine, that the probability, ceteris paribus, is the fame, if the proportion of the number of affirmative witneffes to the number of negative witnefles be the fame ; whereas the probability is to be eftimated by the dif- ference of thefe numbers. WE have already remarked, that in the enacting of a new law, | we ought to have at leaft that probability for the expediency of the law, below which a perfon cannot aét without imprudence. As the manner of determining this degree of probability is ex- tremely [ 1a] tremely ingenious, I cannot avoid mentioning it. The obje& to be attained is equivalent to this, that in the enaéting of a law, the rifk of error fhould not be greater than what we difregard, even where cur own life is in queftion. Buffon and Bernouilli have endeavoured to eftimate the value of this rifk, but the fol- lowing method adopted by Condorcet, feems to be the beft. It is obferved, that from the age of thirty-feven years to forty-feven, and from eighteen to thirty-three, the rifk men run-of dying by accident or difeafes of fhorter duration than a week, encreafes con- tinually in nearly a regular manner; and it is alfo obferved, that a man of thirty-three years is not more apprehenfive of fuch kind of death than a man of eighteen, nor a man of forty-feven than a man of thirty-feven years; the difference of rifk therefore in thefe cafes is difregarded: now, from the tables of mortality, it appears, that, in the firft period, the difference of rifk is = ssirry and in the fecond = +;7';z4; let us then take the latter, which is the greater, as the limit of that rifk which may be dif- regarded, and confequently +4427¢3 will be the limit of the affu- e in th rance, which we ought to have in the enaQing of a new law. Ir we fuppofe that the odds that each legiflator votes juftly, is ten. to one, then will a majority of fix be requifite to give the affurance required ; which in an aflembly of three hundred is only a majority of one in fifty. THESE ek ae Tuese principles, which we have laid down above, may be like- wife applied, as is manifeft, to determine the probability of the de- cifions in courts of appeal; where the fame queftion is fucceflively tried before different tribunals. Anp here [ cannot avoid obferving, that Dr. Waring’s method of determining the refulting probability, where different arguments are contradictory, is erroneous. Let P, fays he, be = the probabi- lity refulting from the arguments in fupport of the fa@, and Q= the probability refulting from all the arguments againft the faa; then the probability of all the arguments for the fa@ will be P—Q, if P be greater than Q,; or againft it = Q—P, if O be greater than P. See Principles of Human Knowledge, § 10. Now, according to thefe principles, if two witneffes of equal veracity fhould con- tradi&t each other, the difference between the probabilities for and againft the fact would be = a, that is, the fa@ would be impoffible ; which evidently cannot, be a true inference. But in reality, in this cafe, there would be an equal chance for the truth and falfhood ‘of the fa&; for let the odds that each witnefs tells truth be the ratio of a to b, then the odds refulting that the fa& is true, will be ‘the ratio of a6 to ba, and the refulting probability = aero ab+ab_ Nj- ° Acatn, if againft a propofition which is abfolutely certain, there fhould occur an argument for the truth of which there was an 4 Vou. VII. P even [ 314 ] even chance; the probability refulting upon the whole, accord- ing to Dr. Waring, would be no more than an even chance, for t—1 = 4; which is manifeftly a falfe inference. In faét, fince the odds that the propofition is true are infinite, or as 1 to o, the refulting odds muft always be as fome finite num- ber to o, that is, infinite, that is, the propofition will ftill be certain. I nave here mentioned fome circumftances relative to the na- ture of the evidence refulting from concurring and contradictory reporters, not tending dire¢tly, it is true,,to the eftablifhment of the point I propofed to myfelf, but nearly connected with it; my principal, and I may almoft fay, my fole object being to fhew, that the evidence of teftimony can overcome any degree of improbability however great, which can be derived from the nature of the fact. Our expectation that a phyfical event, in the courfe of nature, will happen in a particular manner, is founded on previous expe- rience; which experience may be both perfonal and derived; that is. our expectation may be deduced both from our own actual experience, and the reports of others vouching their expe— rience, of the like events. in fimilar cafes. Since this expeClation muti neceffarily be of fome determinate value, depending in fome manner if: rg, manner on the number of experiments either actually made by ourfelves or reported by others, we will fuppofe it = . This ar- gument is founded on an analogy which has never deceived us, and is called, by Mr. Hume, a proof. On the other hand, there is a direct-and pofitive teftimony of a fingle witnefs, that the contradictory of this event did actually happen ; and this is fuch a teftimony as both perfonal and derived experience aflures us has never deceived; the probability of the truth of this teftimony we will call = ; this argument Mr. Hume likewife calls a proof, and he fuppofes, that it is equal to the former, that is, 3 ae : ——= ——. This however is a mere hypothefis; for they are both probable inferences only, deduced from experience; but it is by no means fhewn, that the number of experiments made in both cafes are the fame, or the circumftances exactly parallel; ¢ therefore may be either equal to, or greater, or lefs than ¢, in any _ affigned proportion. The evidence of a fingle witnefs is to be com- ' pared with that probability of an event in phyfical phoenomena, which is derived from a feries of fimilar experiments only; be- caufe the veracity of human teftimony conflitutes one fpecies of events in the courfe of nature, in the fame manner as the finking of lead in water, or the diffolution of gold in aqua regia; and therefore is deduced, in the fame manner as any other fpecific P 2 phoenomenon, [ewa6 | phcenomenon, from experience, and appears to arife, in the fame manner, from an eftablifhed law. This veracity therefore is confirmed by the analogy of other phoenomena, in the fame manner as any given f{pecies of phyfical phcenomena; inafmuch as thefe other phcenomena contribute to eftablifh the general principle, that a// things are conduéted according to eftablifhed laws. If now we confider the numerous experiments we make every day on the veracity of human teftimony in certain circum- ftances, fo that our analogy in this cafe is founded on an inde- finitely greater number of inftances than in any other fpecies of events in the courfe of nature, we may perceive, how the evidence even of a fiagle witnefs may be fo circumftanced, as to eftablifh an individual phyfical phenomenon, however contradictory it may appear to our previous experience of fimilar facts. Let us however fuppofe, that the evidence of the fingle witnefs is lefs than the evidence of experience in any affigned proportion, or that ¢ is lefs than ¢ in the proportion of 1 to m; then mtme, ¢ mt and ANS pT eT Take now fuch a power # of ¢, as that it aa fF e thall be greater that m/, andj will be greater than eae that is, if 2 be the number of witnefles, each of whofe veracity is =77) their concurrent tefltimony will be fufficient to overcome é the probability , Figs derived from the nature of the fact. Hence there‘ore en po therefore it follows, that the evidence of teftimony can approach indefinitely near to certainty ; and can at length exceed the evi- dence of any inference, however cogent, which can poffibly be deduced from perfonal experience, or from perfonal and derived experience conjointly. ’ Ir is to be obferved, however, that the calculation here fated, ap- plies only to the teftimony of different witnefles, who fimply give their evidence as to the truth or falfhood of a propofed fac ; or of witneffes each of whom has an opportunity of knowing what teftimony the others have given. This, without doubt, is to _take the force of concurrent teftimony at the greateft difadvan- tage; meverthelefs, even in this cafe we find, that it has no limit. But there are other cafes in which the leaft number of concur- "rent witnefles, let the degree of their veracity be however fmall, can afford a probability which fhall exceed any given degree of probability however great; namely, where the witnéfles have had no means of knowing each others teftimony, and the fact is at- tended with contingent circumftances, which make a part of their depofition: becaufe the chances of their not concurring in thefe circumftances, may exceed any given chance. In thefe cafes we obferve, that even witnefles who have been obferved to tell falfhood oftener than truth, may yet produce belief; becaufe here the probability of the truth of their report is not derived from the chances of their coinciding, abftrafedly, in truth or falfhood, but from the chances of their coinciding in circum- ftances contingent in their nature, and which have no apparent connetiion with each other. As for inftance, if each witnefs Vou. VIL. Be ae a fhould Me [as J fhould declare, that a celeftial phenomenon, even fuch as we had never feen, had appeared in a certain region of the heavens, ona certain day, hour, and fecond; had run over a_ particular tract, and laftly difappeared with circumftances peculiar and minutely detailed; we muft perceive, that our belief would not be founded on an enquiry into the characters of the witnefles, but folely into the chances of their concurrence in thefe contin-. gent circumftances, [> 3mm, ] On the NUMBER of fhe PRIMITIVE COLORIFIC RAYS iz SOLAR LIGHT. By the Rev. MATTHEW YOUNG, D. D. Gl D.C. De i MR. L.A, Tur opinion that there are but three primitive colours has Read April been maintained by M. du Fay, and after him by Father Cattell. See Montucla, Vol. I. p. 630.; but they and all others who hold the fame doétrine, defend it merely on the principles of a painter, who fhews how with thefe three colours on his pallet, he can compound all others; for with red and yellow he can form an orange colour; with blue and yellow he forms green; and with blue and red he forms indigo and violet; and thus having com- pounded the feven prifmatic colours, it is manifeft that all other colours, with their different gradations, can be formed from them likewife. But this pharmaceutical argument is by no means fufficient to fatisfy us as to the real compofition of folar light. “ LicuT, in refracting, is decompofed into feven rays, red, ‘* orange, yellow, green, blue, indigo and violet. It has been “« {uppofed,” 7th, I 798% [ #1Z0" | fuppofed,” fay Fourcroy, “ that three of thefe colours, the red, yellow and blue, were fimple; and that the other four o n o a * were formed each of its two neighbours; that is, the orange a a from the red and yellow, the green from the yellow and blue, “ the indigo from the blue and violet, and the violet from the “« red and indigo. But this fuppofition has never been proved.” See his Philofophy of Chem. ch. 1. §3. Befides that this is a mere hypothefis, unfupported by any faé&, as Fourcroy obferves, we remark, that it is in itfelfinadequate; 1f, becaufe in the folar fpetrum, the red and indigo are not neighbcuring colours. but are almoft at the greateft poflible diftance from each other. a2dly, According to this hypothefis, indigo is compofed of blue and violet ; but violet is compofed of red and indigo; indigo therefore is compoied of red, blue and indigo, that is, indigo itfelf is one of its own effential ingredients, which is abfurd. Tue experiments of the prifm feem to eftablifh, in a very clear manner, the exiftence of feven original and uncompounded colours; and though green, for inftance, may be compounded of blue and yellow, yet it does not dire@tly follow from thence, that it always is fo actually compounded. Accordingly Newton tells us, that green may be exhibited in two different ways, either by pri- mitive, green-making rays, which are fimple and not refolvable by any reflection or refraction into different rays; or by a com- pofition of blue and yellow rays, which are differently refrangible, and which therefore after their union, may again be fepa- rated a [> san J rated by refraGtion, and exhibit their proper colours of blue and yellow. Ow this do@trine of the two-fold generation of green, we may in the firft place remark, that the antient, received axiom ‘“ Deus “ nil agit fuftra” ought not to be too haftily abandoned, as it mutt appear to be, if this doctrine be maintained: for if green may be produced by blue and yellow, then blue and yellow being already exiftent, green is a confequence; and therefore peculiar rays formed for the production of green are fuperfluous. Though I acknowledge, that this maxim is not fo cogent or felf- evident, as to preclude all objection, yet fince the general obfer- vation of nature feems to fhew, that this wafte of power or multiplicity of means is not adopted by the Supreme Artift, it certainly feems juftly entitled to our attention, at leaft fo far as this, that we fhould be careful in fhewing, that we are led to ; thefe different caufes of the fame effect, by a legitimate and cautious analyfis. In defence of the doétrine of three primitive colours only, F. Caftelli contents himfelf with faying, that the colours of the prifm are immaterial, accidental, artificial, and therefore un- worthy the regard of a philofopher; whereas the colours of painters are fubftantial, natural, palpable. From them, of con- fequence, the theory of chromatics fhould be deduced ; but they Vou. VII. Q tell Ee aa tell us, that there are but three parent colours, which give birth to all others. In reply to this we need only obferve, that Sir I. Newton has proved, that the colours of natural bodies depend on the colorific qualities of the rays of light; and therefore that our theory of colours muft be derived from an enquiry into the conftitution of folar light, for according to that conftitution the colours of bodies will vary: and he farther fhews, that if folar light confifted of but one fort of rays, all bodies in the world would be of the fame colour. However true therefore F. Caftelli’s theory may be, the manner in which he deduces it from phenomena is unqueftion- ably falfe. I fhall therefore proceed to enquire {crupuloufly into the com- pofition of the folar fpeftrum, from which, without doubt, the true doétrine of the origin of colours is to be derived. Ir the folar light confifted of feven primitive, homogeneal co- loured rays, and that thefe homogeneal rays were equally re- frangible, the fpetrum would confift of feven circles of different colours, fince the homogeneal rays of each colour would paint a circular image of the fun. But it is manifeft, that feven circles could not compofe an oblong fpectrum, with redilineal fides. Therefore the rays of the fame denomination of colour mutt be differently refrangible. Which is alfo made ftill farther . evident EES ean a | evident by obfervation of the fpectrum, fince in it we perceive, that the prifmatic colours are diffufed over fpaces, which are, on the fides, terminated by right lines, and therefore the centers of the circles of the fame denomination of colour are diffufed over lines equal to thefe fegments of the reCiilineal fides of the fpectrum. Newton has fhewn, prop. 4. B. 1. Optics, how to feparate from one another theheterogencous rays of compound light, by diminifhing the breadth of the {pe€trum, its length remaining unchanged ; and when the length of the {pectrum is to its breadth, as 72 to 1, the light of the image is feventy-one times lefs compound than the fun’s direct light. In the middle of a black paper he made a round hole, about a fifth or a fixth part of an inch in diameter, upon which he caufed this {fpe&trum fo to fall, that fome part of the light might pafs through the hole of the paper; this tranf- mitted part of the light he refracted with a prifm placed behind the paper, and letting the light fall perpendicularly upon a white paper, he found that the fpeétrum formed by it was perfectly cir- - cular. Hence, therefore, it follows, that the equally refrangible rays occupy a fpace on the rectilineal fides of the fpectrum equal at leaft to the fifth or fixth part of an inch, that is, the rays of the fame colour are differently refrangible. Tue different quantity of the homogeneous rays of different co- lours ‘will not account for the different {paces they occupy in the fpectrum ; for this difference in quantity would afle@t only the intenfity of the colour, not the magnitude of the fpace which it QO2 would fpta4. would occupy. All the red light therefore is not homogeneous ; but confifts of rays of innumerable, different degrees of ae gibility ; and fo of the other colours. Now fince the rays which are of the fame denomination of co- lour are differently refrangible, they will either form oblong fpectrums detached from each other ; or they will in part lap over, and fall on each other. The former pofition is manifeftly falfe: therefore the original prifmatic colours will partly lap over and fall on each other, and therefore neceflarily generate the interme- diate colours. And fo Sir I. Newton obferves, where he fays, that the original, prifmatic colours will not be difturbed by the intermixture of the conterminous rays, which are intermixed together. This overlapping however, which Newton {peaks of, arifes only from the fun’s having a fenfible diameter, and does not neceflarily impiy an equal refrangibility in any differently co- Joured rays. If there be but three original prifmatic colours, red, yellow and blue, and that the red and yellow lap over, fo as that there fhall be a certain fpace in the fides of the fpeCtrum equally occupied by yellow and red circles, then will thefe circles by their intermixture compound an orange colour; and this co- lour as to refrangibility will be homogeneous, becaufe the coin- cident rays of different colours are equally refrangible. In like manner green may be compounded by the mixture of blue and yellow circles, equally refrangible. Now this is fimple, and con- formable to the other phenomena of the {pectrum; for if rays of the bas. the fame denomination of colour be differently refrangible, it 1s not unreafonable to fuppofe, that rays of a different denomination of colour may be equally refrangible; and therefore fince the red tays are unequally refrangible, and likewife the yellow, there is nothing incongruous in fuppofing that fome of the lefs refrangible of the yellow may be equally refrangible with fome of the more ‘refrangible of the red; and if fo, they will confequently be in- termixed with them: and the fame may be faid of the green. This hypothefis likewife receives confiderable ftrength from this confideration, that the orange, green, indigo and violet oc- cupy thofe places which they ought to do, in cafe there were but three primitive colours, red, yellow and blue: thus the orange lies between the red and yellow, becaufe it is formed by fome of the extreme rays of red and yellow, which are equally refrangible; in like manner the green lies between the blue and © yellow, becaufe it is formed by the mixture of blue and yellow. The indigo and violet muft alfo occupy the extreme part of the fpe€trum, where the moft refrangible red and blue rays are united, and gradually becoming more and more dilute, fade away, and at length entirely vanifh. But if the orange, green, indigo and violet be primitive colours, there is no apparent reafon why they fhould have had fuch degrees of refrangibility afligned them, as that they fhould occupy the places they do, rather than any. other. Moreover, if thefe three colours red, yellow and blue.be the primitive colours, they cannot themfelves be generated; and ac- cordingly i 36 -| cordingly we find, that yellow cannot be generated by the mixture of the adjacent prifmatic colours, orange and green ; and the reafon of this is evident, becaufe orange is compounded of red and yellow ; and green is compounded of yellow and blue; but red and blue compofe purple; which added to the yellow will generate a new compound colour, viz. a fickly green, differing manifeftly from yellow, the colour which ought to refult according to the analogy of the other primitive colours, in which the extremes, by their mixture, generate that which is intermediate. In the fame manner, blue cannot be generated by the mixture of green and indigo, becaufe green is compofed of yellow and blue, and indigo of blue and violet; therefore the refulting colour is compofed of blue, yellow and violet ; but yellow and violet do not compofe blue, therefore neither will blue, yellow and violet compofe a blue colour. Now if orange and green be primitive colours, in the fame manner as red, yellow and blue, we can affign no reafon why blue fhould not be generated by the mixture of the adjacent colours, as well as green and orange. But it is a received prin- ciple, that an hypothefis fhould folve all the phanomena; of the two hypothefes therefore, namely, thatthere are feven primitive colours, differently refrangible ; or that there are but three, fome of which, of each fpecies, are equally refrangible ; the latter alone folves all the phenomena of the folar fpectrum, and therefore is to be preferred. Ir it be faid, that thofe rays which are equally refrangible muft excite the fame fenfation on the retina, becaufe they muft have [Svrege | have the fame momentum; it is replied, 1ft, That it has not yet been proved, that the fenfation of different colours depends on the different momentum of the rays. 2dly, The rays may have different momentums, and yet be equally refrangible; for fince refraction is fuppofed to depend on the attractive force of the denfer medium, we mutt fuppofe it analogous to the attractive force of gravity, which is proportional to the quantity of matter ; and therefore the greater or lef{s quantity of matter in a particle of light would produce no alteration in its refraction. Neither can the different refrangibility depend on the different velocity of the rays; becaufe the difference of refrangibility of the ‘red dnd violet rays is much greater in flint glafs than in crown glafs ; and this would require a proportionably greater difference in the original velocities, which cannot be. And this fame argu- ment holds equally againft the former hypothefis, that the diffe- rence of refrangibility depends on the different magnitude or _ denfity of the particles of light. 3dly, Refraction feems to arife from a fpecies of eleCtive attraction, fince different mediums which act on the mean rays equally, act on the extreme rays unequally: hence rays of the fame quantity of matter and ve- locity, and therefore of the fame momentnm, may be diverfely refracted ; and rays of different momentums equally refracted. Nor is it to be wondered at that the rays of light fhould be differently refrangible, independent of any regard to their mo- mentum, when we confider, that the different coloured rays ap- pear fe 128s | pear to be combined with combuftible bodies, with different degrees of attractive force. For in combuftion we find, that different bodies are difpofed to part with different rays with greater facility ; but when the combuttion is fufficiently rapid, they part with all the different coloured rays together, and the flame is therefore white; and this is what is called a white heat. Dr. Fordyce in the Phil. Tranf. for 1776, tells us, that when the heated fubftances are colourlefs, they firft emit a red light; then a red mixed with yellow, and laftly, with a great degree of heat, a pure white. All this is wonderfully conformable to the refraction of light by tranfparent fubftances, which refract, and therefore at- tract the red light lefs, and confequently in combuftion part with it more eafily. On the other hand I know it is generally believed, that the light in combuftion proceeds from the air, but this circum- {tance of the different colour of the light in different cafes, feems to overturn this opinion; for if vital air were oxygen diffolved in caloric and light, then the oxygen being abforbed by the burning body, the light extricated would in all cafes be of the fame nature ; the greater or lefs rapidity of the combuftion would only | produce an extrication of a greater or lefs quantity of light, but could not produce any variation in its nature, it being neceffarily the fame in all cafes, to wit, that in which vital air is diflolved. But the truth or falfhood of this reafoning will not affect the validity of the pofition, that the refrangibility of the rays of light cannot depend on the different magnitude, denfity or velocity of the particles. Bur [tag 3 | Bur though fpeculation feems thus to render it probable, that there are but three parent colours ; e theory muft ever remain unfatisfactory, unlefs it receives the fanction of dire& experiment. In this however there is no {mall difficulty ; for fince the rays of light which compofe any given individual point of the colours of orange, green, violet, and indigo are equally refrangible, they will be alfo equally reflexible; and therefore cannot be feparated either by refraction or reflection, fo as to exhibit the different coloured rays of which they are compofed. It feems therefore, that the only way remaining, by which we can experimentally afcertain the compofition of thefe colours, if they be indeed compound, is tranfmiffion. For fince tranfparent coloured bodies are fuch merely by their letting pafs through them either folely, or more copioufly, rays of a certain colour, and intercepting all others, fuch tranfparent bodies, applied to compound colours, will afcertain that compofition, by extinguifhing, in a great mea- ” fure, all rays except fuch as are fo adapted to its conformation, as to pafs through it, and give it its peculiar denomination of colour. - In order to try the truth of the hypothefis of feven colours by _ this teft, I looked through a blue glafs at the red end of the _ fpe@rum : now we are to confider, that if that part of the fpectrum : * was compofed of red rays, and none other, the only effe& of the ® blue glafs would either be a total or partial fuffocation of the red ‘fays; and therefore that ass of the fpedtrum, when looked at Vou. VII. R ee pean *f through the glafs, would either totally difappear, or become a faint and diluted red. But, on eMfferiment it appeared of a purple co- lour. The purple in this cafe could not be a primitive and ori- ginal colour, as is manifeft, becaufe it did not proceed from the purple part of the fpe@rum; we muft therefore conclude, that it was a compound colour. But purple, when compound, is made up of blue and red, therefore it follows, that fome blue rays did actually exift in the red part of the fpe@rum; which combined with the few, ftraggling red rays which penetrated the blue -glafs, compofed that purple colour, which the red extremity of the fpe@trum affumed, when viewed by the light tranfmitted through the blue medium. To try, on the other hand, whether any red rays lay hid amongft the blue, I proceeded in the fame manner, and looking at the blueft part of the fpeGrum through a red glafs, it appeared of a purple colour; fome red rays therefore are equally refrangible with the blue; and if the red extends as far as the blue, there is no reafon why we may not fuppofe that it extends fomewhat far- ther, fo as to compound, with a diluted blue, the extreme colours of the fpetrum, indigo and violet. Bout it may be faid, that if blue rays exifted amongft the red, ~— that part of the fpe€trum could not appear fo extremely brilliant as it really does; but would put on a purplifh appearance in the fpectrum itfelf, even to the naked eye. In anfwer to this objection al we. \ al [ 13: ] we may obferve, that the moft intenfe and vivid, natural red bo- dies do, in fact, reflect a very great proportion of blue rays, be- caufe they appear of a ftrong blue colour when placed in the blue part of the fpectrum; and therefore they refle&t juft as many when the dire&t, white folar light falls on them, in which all that blue is involved ; though by the predominance of the red rays, they appear of that colour, without any vifible tin@ure of blue. In order to determine whether the purple appearance of the red extremity of the fpecrum, when viewed through a blue glafs, was caufed by any of the white folar light, which might perhaps be refleCted from the air, or furrounding obje@ts to the fpe@trum, and thus throw on that part fuch a quantity of blue as might produce a fenfible effect; I caufed the middle and moft intenfe part of the red to pafs through a hole in a blackened paper, and : then fall on an optical fereen; by which I was fure that I had as pure and uncompounded a red as could be defired ; which alfo un- derwent the ufual teft of purity by fubfequent refra@ion, without any change in the form of the fpecrum; I then looked at the body which was i!luminated with this red, through the fame blue glafs, and the effet was the fame as before. To try this doGrine of three parent colours ftill farther, I con- fidered, that if the orange were really compounded of the red and yellow rays, then by looking at the orange through a red glafs, 2 R 2 _ the [erga | the orange would in a great meafure vanifh, and the red would appear to extend much farther than in the original fpe@rum ; be- caufe the yellow rays being confiderably obftruted, the red would become more predominant ; and that part of the fpe€trum, which before appeared orange, in confequence of a certaim mixture of yellow and red, would now, by the failure of fo confiderable a part of the yellow, lofe its orange appearance, and put on that of red: and, on experiment, I found the cafe to be fo really in fa@; for while an affiftant looked at the fpe@trum through the red glafs, I moved an obftacle from the red towards the other end of the fpeftrum, defiring him to ftop me, when the obflacle fhould arrive at the confines of red and-orange ; but when he did fo, the obftacle had attained the middle of the orange, or rather had paffed beyond it. Now if the orange were really a primitive colour, I fhould fuppofe, that when looked at through the red glafs, it would either appear diluted, without any change of dimenfions ; or that if the weak part of the orange, next the red, fhould va- nifh, by the obftrudtion of the glafs, a dark interval would appear between the orange and the red; in neither cafe can we account for the apparent extenfion of the red into the region of the orange; nor by any other hypothefis, as appears to me, than that fome of the red rays are equally refrangible with fome of the orange. Tuere is another argument derived from the ocular fpe@tra of Dr. Darwin, which ftill further corroborates the doétrine of three primogggial Cote preset J primogenial colours. Place a piece of coloured filk, about an inch in diameter, on a fheet of white paper, about half a yard from your eyes; look fteadily upon it for a minute; then remove your eyes upon another part of the white paper, and a fpe@trum will be feen of the form of the filk thus infpe@ted, but of a diffe- rent colour, thus Red filk produced a green fpe@trum, Green - red, Orange - blue, Blue - orange, i Yellow - violet, Violet - yellow. Tue reafon of thefe phenomena is very ingenioufly affigned by” Dr. Darwin; he fays, that the retina being excited into a violent and long continued action by the red rays, in the firft experiment,, at length is fo fatigued as to become infenfible to them; but that | it ftill remains fenfible, that is, liable to be excited into ation by any other colours at the fame time; and therefore the fpeQrum affumes a green appearance, becaufe if all the red rays be taken out of the folar light, the remaining rays will compofe green. See Phil. Tranf. Vol. LXXVI. Converfely, a green obje@ produces a red ocular fpeGrum.. Now we may obferve, that if all the green: rays be taken out of the folar fpe€trum of feven colours, the re-- maining colours will not compound red. If indeed green be not: a. Beaetr colour,, but a compofition of blue and yellow, then . will: L e734: | will the eye, in looking on a green objet, be at once affected by blue and yellow rays; and therefore become infenfible to them both; and confequently the fpe@rum will appear red. But’ if green be a primitive, original colour, generated by its own peculiar green-making rays, the eye in contemplating a green object, will become infenfible only to the green rays; and therefore the other fix prifmatic colours, which are fpecifically different from the green, ought to be fenfible, and produce their proper compound effet; but this would not be the fenfation of red. In like manner, if the object be yellow, the eye will at length become infenfible to the yellow-making rays, and the fpectrum will be violet. Now fince on the hypothefis of feven original colours, the orange and green are primitive, though the eye be rendered infenfible to the yellow rays, it will not be fo to the orange and green, which therefore, together with the red, blue, violet and indigo will produce their compound effect; but the colour re- fulting from this joint action is not violet, which neverthelefs is the colour of the ocular fpetrum. On the other hand, if there be but three primitive colours, red, yellow and blue, when the eye is infenfible to the yellow-making rays, the fpe@trum muft neceffarily be violet, which is the colour that refults from the mixture of red and blue. If it be objected, that the eye is not only infenfible to the unmixed yellow rays, but likewife to the ycllow of the orange and the green, then it is admitted that orange and green are compound colours. Eefides, fincg the co- lour which would refult from the mixture of red, orange, green, blue, * iss 4 » blue, indigo and violet is not yellow, the eye ought not to be in- fenfible to this colour; and confequently, fince by the exemption of the yellow rays from the white folar light, that colour does not refult, but a diflincét purple, it follows, that the orange and green are not primitive colours inherent in folar light. Ir remains now only for us to fhew, that the three colours of red, yellow and blue are adequate to the folution of all the phe- nomena of chromatics. But in order to fhew this, few words will be fufficient, for having feen, that the feven prifmatic co- lours can be generated by thefe three, it follows that all others can be generated from them, as Sir I. Newton has proved at large. However I think it will not be fuperfluous to. obferve, that: white may be direétly produced by thefe three colours, without: the previous generation of the other four prifmatic colours, in the fame manner as it is ufually generated with feven. “ I'could: “ never yet,” fays Newton, “ by mixing only two primary co- ** lours, produce a perfect white. Whether it may be compofed “ of a mixture of three, taken at equal diftances in the circum-- ** ference, I do not know.” Now to fhew that white may be- thus generated, let an annulus of about four inches diameter be- divided into three parts by lines tending towards the centre, and let thefe three divifions be refpectively painted red, yellow and. blue, in proportions to be afcertained by trial; then if the annulus be turned fwiftly round its centre, it will appear white. That white may be generated by the mixture of. only the three: colours ; @a139 ) colours red, yellow and blue might alfo appear from the rule. which Newton himfelf has given us, for determining the colour of the compound which refults from the mixture of any primary colours, the quantity and quality of each being given. Tue rule is this, the circumference of a circle is diftinguifhed “into feven arches proportional to the feven mufical intervals in an octave, that is, proportional to the numbers 45, 27, 48, 60, Ge. 4c, 80: the firft part is to reprefent a red colour, the fecond orange, the third yellow, the fourth green, the fifth blue, the fixth indigo, and the feventh violet. Thefe are to be confidered to be all the colours of uncompounded light gradually paffing into one another, as they do when made by prifms, the circumference reprefenting the whole feries of colours from one end of the fun’s coloured image to the other. Round the centers of gravity of thefe arches Ict circles proportional to the number of rays of each colour in the given mixture be defcribed. Find the common centre of gra- vity of all thefe circles, and if this common centre of gravity coin- cide with the centre of the circle, Newton fays that the com- pound will be white. Join therefore the centers of gravity of the blue and yellow circles, and from the centre of the red circle draw a right line through the centre of the principal circle; from the conftruction it will cut the line which joins the centers of the blue and yellow circles; if therefore the number of the blue and yellow rays be to each other inverfely as their diftances from the point where the line which joins their centers is cut by the line drawn from the centre ike aa centre of the red circle; and if the number of red rays be to the fum of the yellow and blue fays inverfely as the diftances of the centre of the red circle, and the common centre of the yellow and blue from the centre of the principal circle, the common centre of gravity of the red, blue and yellow circles will coincide with the centre of the principal circle, and therefore the refulting compound ” will be white. Bor it is manifeft that this conftruction cannot be relied on, becaufe the quantities of the rays of any given colour in folar light, do not appear to be proportional to the fpaces which they occupy in the retilineal fides of the fpeGrum. Thus it is known that the yellow making rays are predominant in folar light, yet the fpace they occupy in the fpedtrum is to the fpace occupied ei-. ther by green or blue as four to five, and to the fpace occupied by the violet only as three to five.. Vou. VII. S: OBSERVATIONS on the THEORY of ELECTRIC ATTRAC- TION and REPULSION. By the Rev. GEORGE MILLER, Pee GD, ee that the theory of a fingle eleGric fluid was propofed, nevis sth no difficulty occurred in the explanation of the attraGtions and re- pulfions obferved to arife from eleGtricity. If we admit that there are two diftin@ eledtric fluids, each of which ftrongly attraéts the other, ‘but confifts of particles mutuaily repulfive; it becomes eafy to account for the attraclion fubfifting between bodies in different ftates of ele@ricity, and the repulfion between thofe in the fame. But when Dr. Franklin*, obferving that a man, ftanding upon a non-conduétor, could not electrify himfelf, but that he could electrify another perfon alfo ftanding upon a non-conduétor, was induced to regard the operation of exciting electricity only as a transfer of one and the fame fluid from one body to another; it was found to be difficult to reconcile to the new theory the mutual _repulfion of bodies in that ftate which is, according to this theory, 82 denominated * Dr. Prieftley’s Hiftory of Ele€tricity, p. 161. [. 40 ] denominated negative eleCtricity. Dodtor Franklin* acknowledged” that he could not affign a fatisfa@ory reafon for it; and Doctor Prieftley | has propofed it, as one of the queries remaining to be folved for completing the fcience of eleGricity. Many attempts have been made to obviate this apparent objection to the fimple theory of a fingle fluid; but the difficulty feems ftill to be as great ds it was in the time of Franklin. { Zrrnus has applied a very elaborate fyftem of mathematical reafoning to the folution of electrical phenomena, and has adopted as the bafis of his theory, the fame opinion which Franklin: had entertained concerning the nature of the eleétric fluid; but he has combined with this opinion other principles fo in- admiffible, that his reafonings cannot be regarded as juft. expli- cations of the phenomena. He has affumed, apparently without any other reafon than its importance to his conclufions, that the particles of all other fubftances repel each other. His fyftem muft therefore be confidered, not as a phyfical folution agreeable to the known laws of natural operations, but merely as an ingenious exercife of mathematical ability. M. De Luc, who rejected the folutions of Zpinus has endea- voured to fupply the deficiency. § Having remarked that the divergence * Dr. Prieftley’s Hiftory of Electricity, p- 165. + Ibid. p. 492. t Journal de Phyfique, Dec. 1787. § Journal de Phyfique, Juin 1790. © Re ea ae 9 divergence of the balls of an ele€trometer, included in the receiver of an air-pump, is continually diminifhed during the progrefs of exhauftion; he confiders it as proved, that the caufe of all elec- trical movements, whether of attraétion or of repulfion, is the aGiion of the air. This principle he applies in the following manner. When two bodies are in fimilar ftates of ele€tricity, ei- ther pofitive or negative, they will confpire to modify, either by giving or receiving the eledtric fluid, the ftate of the intermediate air, whilft that of the exterior air is only modified by either of them fingly; and therefore the ftate of the exterior air will differ more from that of the electrified bodies, than the ftate of the in- termediate air. In this cafe he contends that a repulfion muft take place, becaufe each body muft move towards that part of the furrounding medium, whofe eletrical ftate is moft different from its own. On the other hand, when bodies are in different ftates of elericity, they will mutually countera& the changes, ‘which they might feparately produce in the ftate of the intermediate air; but each will operate on the exterior air without any compenfation. In this cafe the ftate of the intermediate air will continue puditiet from that of each body as much as at the firft inftant, whilft the ftate of the exterior air is feparately modified by each body accord- ing to its refpective ftate of eleCtricity. The two bodies therefore, moving towards that part of the furrounding medium, whofe elec- trical ftate is moft different from their own, will at the fame time move towards each other. THIS [ yee 7] Turis theory very ingenioufly avoids the difficulty of explaining the cafe of ele@rical repulfion, by refolving it into an attra@tion towards the furrounding medium. It feems however to be liable to two objections. In the firft place, inftead of affuming unau- thorized principles with the preceding theory, it omits the confide- ration of one whofe exiftence feems to be afcertained by experi- ments. If a body be in cither ftate of electricity, it will induce in an adjacent body the contrary ftate, until it fhall have come within a certain diftance. This property, which has been afcer- tained by various experiments, indicates a repulfive force fubfifting between the portions of the eletric fluid that belong to the adjacent bodies; and this theory makes no allowance for fuch a repulfion. The fundamental principle of it is merely a diffufion of the electric fluid, and is * thus ftated by M. De Luc: “ the electric matter tends towards all fubftances, and the more ftrongly in the fame propor- tion in which they poffefs a {maller quantity.” In the fecond place, it does not appear, when carefully confidered, to afford any affift- ance towards the removal of the grand difficulty, the mutual repul- fion of bodies negatively electrified. If two bodies negatively elec- trified be placed at a fmall diftance, they will both, according to M. De Luc’s explanation, receive the electric fluid from the inter- mediate air, which will confequently retain a fmaller portion than the furrounding atmofphere From the law above-mentioned it fhouid follow, that the redundant fluid of the exterior air fhould by 2 diffufion * « La loi fuivante fufit feule: La matiére éleétrique tend vers toutes les fub- « ftances, d’autant plus fortement, qu’elles en pofledent moins.” Journal de Phyfique, Juin 1790. PKs. oc] diffufion be communicated both to the bodies and to the inter- mediate fpace; but no reafon appears, which would induce us to fuppofe that the bodies themfelves fhould recede to a greater diftance. M. De Luc does indeed endeavour to prove that fuch a motion fhould take place, but by an experiment whofe folution contradiéts his own theory. He fufpended by a filk thread a large, but light, metallic ball, and prefented it in a ftate of pofitive elec- tricity to a body negatively electrified. The former was attracted towards the latter until it arrived at a certain diftance, at which it difcharged its ele@ricity. Hence he concluded, in general, that when a body has more of the ele@ric fluid than the neighbouring bodies, and is lefs difpofed to refift its own motion than to abandon the excefs of its eleftric matter, it will move towards that place which contains lefs of this matter. But in this experiment he con- fiders the two bodies as acting on each other at a diftance without any reference to the intermediate air. - Mr. Cavauto*, in the laft edition of his treatife on electricity, has obferved, that the mutual repulfion of two bodies negatively ele@trified is ftill fuppofed to contradi& the theory of Franklin; and has therefore deemed it neceffary to obviate the objeétion by a very particular detail. For this purpofe he has premifed the following propofitions. Prop. 1. No electricity can appear on the furface of a body, or no body can be eletrified either pofitively or negatively, * Vol. Ill. p. 192. [ “344 ] negatively, unlefs the contrary electricity can take place on other bodies contiguous to it. Prop. 2. There is fomething on the fur- face of bodies, which prevents the fudden incorporation of the two eleCtricities, viz. of that poffeffed by the ele€trified body with the contrary eledtricity poffeffed by the contiguous air, or other furrounding bodies. Prop. 3. Suppofing that every particle of a fluid has an attra€tion towards every particle of a folid; if the folid be left at liberty in a certain quantity of that fluid, it will be attra@ed towards the common centre of attraCtion of all the par- ticles of the fluid. To this laft propofition he has fubjoined the two following corollaries: 1.* the fame thing muft happen, when the quantity of fluid is {maller than the bulk of the body; 2. if the attraction of the particles of the fluid be exerted only towards the furface of the folid, the effet will be the fame when the body is of a regular fhape; but the difference will in any cafe be inconfi- derable. Wirs regard to the folution founded upon thefe principles it muft be remarked, that it is not derived fimply from a confidera- tion of the fuppofed nature of the electric fluid; but from a mixed ftlatement of that nature and of properties affumed merely from experiments as matters of fat. The firft and fecond propofitions exprefs thofe properties, and, though the experiments to which the former refers, may be explained by afcribing the phenomena to the repulfive nature of the fluid, yet the latter is affumed ‘** without * OF this corollary Mr. Cavallo does not appear to make any diftint application. Ets: J -without any fuch reference. “ Without examining,” fays Mr. Cavallo, “ the nature, the extent, and the laws of this property in bodies, ** it will be fufficient for the prefent purpofe to obferve, that the *“* fa&t is certainly fo; for otherwife a body could not poffibly be “* electrified, or it would not remain electrified for a fingle moment.” From thefe principles thus affumed, Mr. Cavallo deduces the exift- ence of atmofpheres of contrary electricity exifting in the air conti- guous to the bodies; and from the attractions which are thereby occafioned he infers the apparent repulfion of the eledétrified bodies. Ir thefe atmofpheres be conceived to be formed by the repulfive nature of the fluid, fome allowance fhould be made for the mutual repulfion of the two redundant portions belonging to bodies pofi- tively electrified. This however feems to be negleéted for the purpofe of explaining the repulfion of bodies negatively electrified. But the difficulty feems to be only changed. If the negative at- mofphere adjacent to a body pofitively electrified be caufed by the repulfion of the redundant fluid of the body, it will be neceffary to fhew that this repulfion is overpowered by the attra@tion fubfift- ing between that redundant fluid and the portion of air thus de- prived of a part of its eledtric fluid. Bur the reality of thefe atmofpheres of contrary electricity may well be queftioned. It feems to require, that we fhould conceive a portion of air contiguous to each body to be permanently, during Vou. VII. T -- the [. M6] the mutual repulfion of the bodies, in a ftate of electricity oppofite to that of the bodies. But * itis afcertained experimentally, that the air furrounding any electrified body acquires the fame electricity which had been pofleffed by the body, and retains it even after the removal of the body. This muft be fuppofed, agreeably to the known laws of ele¢tricity, to be communicated by the alter- nate attraction and repulfion of the adjacent particles of air. Each particle muft be firft attraéted towards the body, and, when by contact it has acquired the electricity of the body, re- pelled from it. Inftead therefore of a permanent ftate of contrary electricity conflituting thefe fuppofed atmofpheres, each adjacent {pace muft be occupied by particles, fome of which are attracted and others repelled. The time requifite for thus reducing the electricity of the body to an equilibrium with that of the fur- rounding air, is fuflicient for explaining the continnance of the electricity of the bodies, without the aid of the fecond propofition ; and the firft propofition is deduced only from a confideration of bodies in a folid ftate. Possrsty a more diftinct application of a principle, already in fome degree adopted both by Doétor Priefiley and Mr. Cavallo, may remove all the difficulties of this inquiry. At leaf I will hope, that it may lead to fuch a confideration of the queftion, as may fubje@t the merits of the theory itfeif to a fair and decifive difcuffion. * Cavallo’s Complete Treatife on Ele€tricity, Vol. I. p. 326. cb ae] difcuffion. This principle is faturation. * Doétor Prieftley has ex- plained the communication of the redundant fluid of a body pofi- tively electrified to another, a part of whofe fluid had been previ- oufly expelled, by fuppofing that it was more ftrongly attracted by the other body, than by its own which had more than its natural fhare; and } Mr. Cavallo has in the fame manner accounted for the mutual attraction of bodies in different ftates of e!ectricity. I applying this principle to the folution of eleétric pheno- mena three forces muft be confidered: 1ft, the attraction fubfitt- ing between each body and its own portion of the ele@ric fluid; adly, the attraction which may fubfift between each body and the portion of fluid belonging to the other; and 3dly, the repulfion fubfifting between the two portions of the electric fluid.. Tuat the attraction fubfifting between two bodies in oppofite fates of electricity may be explained, it is neceflary to confider previoutfly the cafe of two bodies in their natural or ordinary ftate. In this cafe the force fubfifting between each body and its own portion of the eleétric fluid is not in a ftate of faturation, becaufe it muft be fufficiently {trong to counterbalance the elafticity of the fluid. Each body is therefore ftill capable of being attracted by the fluid belonging to the other, and each portion of the fluid is alfo capable of fuch attraction. This force, if it fhould operate aid Get” alone, * Hiftory of Electricity, p. 253. + Vol. I. p. 109. [ 145 ] alone, would draw the bodies together; but the mutual repulfion of the two portions of the fluid tends to produce the oppofite effect. The quiefcence of the bodies proves the equality of thefe forces. Ir two bodies in oppofite ftates of eleCtricity be brought together, the body pofitively electrified cannot be attracted towards the remaining ele€tric fluid belonging to the other, becaufe this body may be confidered as faturated with the fluid, and that portion of the fluid as faturated with folid matter. For the oppofite reafons an attraction will take place between the body negatively electrified and the fluid belonging to the former. It remains to be fhewn, that this attractive force may exceed the mutual repulfion of the two portions of fluid. It muft be obferved, that the repulfion re- mains the fame, becaufe the fum of the two quantities of fluid is not altered; whereas the attraction is augmented by the unequal dif- tribution of the fluid. The one body is charged with more fluid than that which its own attracting force is capable of retaining, and the redundant fluid will confequently be ftrongly impelled towards the other body, whofe attractive power is at the fame time increafed by the deficiency of its own portion of fluid. In the cafe of two bodies fimilarly electrified the bodies may be either both pofitively, or both negatively electrified. When they are both pofitively electrified, they are both faturated with the eleciric fluid; and when they are both negatively eleCtrified, both ppedo | both remaining portions of the ele€tric fluid are reciprocally faturated with folid matter. In neither cafe therefore can any attration take place between either body and the fluid belonging to the other. Confequently, the repulfion exifting between the two portions of the fluid muft operate without refiftance, and the two bodies be repelled from each other. SHovuLp this folution of cleric attra€tion and repulfion be ad- mitted, it will perhaps alfo remove the difficulty of magnetic re- pulfion. In this part of philofophy it has been found difficult to explain the repulfion of the correfponding poles agreeably to the theory ofa magnetic fluid. In every magnetical body the equilibrium of this fluid is fuppofed to be difturbed, and one part of the body is conceived to be overcharged with the fluid, whilft the other is undercharged. The difficulty was to explain the repulfion of the undercharged poles, as in electricity to explain the repulfion of | bodies negatively electrified. Mr. Kirwan has indeed, in a Me- moir contained in the Sixth Volume of the Tranfaétions of the "Academy, referred the phenomena of magnetifm to cryftallization ; - but his mention of the term /aturated in that Memoir feems to imply, that he does not mean to exclude the fuppofition of a mag- netic fluid. If this be adopted, the preceding folution may be ap- plied to the phanomena of magnetifm, in the fame manner in which it has been already applied to thofe of ele€tricity. Tue theory, according to which the preceding folution has been propofed, fuppofes the electric fluid a fing/e fluid; but it is not ne- ceffary : lips a ceffary that it fhould be conceived to be abfojutely jmp. We know, for inftance, that atmofpheric air is a combination of at leaft two diftin@ fluids; and yet explain the phenomena of the barometer, air-pump, and condenfer, as depending merely on its prefence or abfence, without any reference to the compofition of its nature. In the fame manner fome ele@tric phenomena may be juftly explained by confidering them as the effects of the diffe- rent diftribution of the fame fluid; whilft its phofphoric fmell, its power of changing blue vegetable colours to red, and its combuftion may poflibly be derived from its decompofition. [ apr J 4 GENERAL DEMONSTRATION of the PROPERTY of the CIRCLE difcovered by Mr. COTES deduced from the CIRCLE only. By the Rev. J. BRINKLEY, 4 M. ANDREWS Pro- Selfor of Aftronomy, and M.R.L. A. My HE very elegant property of the circle difcovered by the cele- brated Cotes has for its extenfive ufes always been juftly efteemed among mathematicians. ‘The inventor left no demonftration of it; and although it immediately excited the attention of the moft abamaknt. cultivators of the fcience, yet no general invefligation has been hitherto given, if we except one derived from ine hyperbola and impoflible expreflions, which was firft given by De Moivre, afterwards by Maclaurin and other authors. But the elegance of the theorem and the ftridtnefs of mathematical reafoning feem to require a very different kind of demonftration. The author of. “ Epiftola ad Amicum de inventis Cotefii,” has indeed attempted a demonftration from the circle only ; however it will readily appear on examination that it is not general, even conceding the demon- ; {tration Read Noy. 4th 1797- [ 152 ] ftration of the theorem for expreffing the cofine of a multiple arc in terms of the cofine of the fimple arc. No author before Dr. Waring has given a general demonftration of this latter theorem, and confequently all demonftrations of Cotes’s property by the circle alone previous to his, cannot be general fo far as that theorem is concerned, and it will be found that in another circum- ftance not lefs important they are all defe@iive. Dr. Waring in his letter to Dr. Powell has from his theorem for the chords of the fupplement of a multiple arc fhewn the truth of Cotes’s pro- perty in particular inftances, and in his “ Propr. Algebr. Curv. Prob. 32,” has given the heads of a general folution. But it ap- pears one of the fteps there omitted is the only difficult part of the demonftration after conceding the theorem for the cofine of a multiple arc. Tue demonftration here given is general and probably as dire& and fimple as the propofition will admit of. The proof of the lemma which it was neceffary to premife is much the moft difficult part of the whole, and it is in that ftep of the de- monftration where the Lemma is applied that all demonftrations heretofore have been defeétive and only applicable to particular inftances, ‘Lemma. [ 153 ] Lemma. Ir m and z reprefent any affirmative whole numbers: then — —=— +i N—l.2—2.9—3. - = = 2 A—M—I yy! i meme gs pe am +: 1m—3. Aad n—5. -~ - - - n—m-+ 1 Xx m. M—I — —————____ +: n—mti.n—mt2 - .- = n—2m—IX1 Wuere 1, m, m.m—t1, &c. are formed by the law of the 2 coefficients of a binomial raifed to the m+. power. The number er ferms = w-+-1. Demonstration. Let the terms of the annexed table repre- fent the different expreffions for the above quantity, according to the different values of m and z. Te ACB. La B Se Values of n : { Sot Mi: 2 37 .. (n—2n—I] on Zs Ao sie * . Se 3 |p A BC, ae : 0 Se apes ah} ”> m—I | Pris +a H K Bs | = | — | | — | — — z i im Se al | POS I TOS a «Mis Vor. VII. U THEN L @s4- J THEN I. By fubftituting x—r inftead of 7 in the above [+ M—2, MN—3 - - - N—MXI —— ————— we have K‘ = 3 | L &c. - &c. therefore (+u—2.—-3 - - XI ee NW—3.A—-4 - - Xm (et eat | De ae Ce (enh: x m by ces ai eee: But by fubftituting ~—2 and 7—! ref{peCtively inftead m—1 inftead of m we have expreffion 1 #3. 2h -- = AM +X Mm bt nh. 5 ~~ mt ax MMI of 7 m, and r+ 2—3.mu—4 - XI +2—2.N—3-xI Ba) em - X m—!I and K = —N—3.N—4 - XM—TI &c. &c. &c. &c. or H+ K y mt = LD—K or Lam—1 XH+ K+ KY 2. Taking m=z the expreflion becomes 4 pt ne em et EX = 7 —7—2. N—3 - - - 7 «LKOXIM | - - - am = = 4 C yA RR Sie SM Coons ae, ia a tae he ae el BE 1) Wuicu L 155 |] Warcs will be = 0, becaufe the firft and laft terms are the fame with contrary figns, and becaufe o will be a factor in each of the other terms. That the firft and laft terms will have cons trary figns appears from confidering that in the laft term there &. are m—I negative factors, and confequently when z is even the product will be negative and the fign of the term itfelf will be pofitive becaufe m+ 1 (m+ 1) is odd, and when z is odd the produc will be pofitive and the fign of the term negative. 3. SusstTiTuTING for m, 2, the general term of the firft hori- zontal rank = + 2—1 : =0 From thefe different conclufions we collect: rift, that (be- caufe ‘L = m—1. H4K-+Ky’) if each of the terms in any hori- zontal rank = o the terms in the rank below are equal: adly, therefore it follows becaufe a term in each rank = o (when m=n) that if each of the terms in any horizontal rank are equal to o, that the terms of the rank beneath are each = 0, and 3dly, becaufe thofe of the firft horizontal rank are each = 0, it follows therefore that each term of the table=o. Q,E.D. THEOREM. 1. Ler the circumference of a circle be divided into x equal ‘parts OO’, O'O", &c. and from a point P in the radius OC or U 2 the Pe gh. | a , cd the radius produced without the circle draw PO, PO’, &c. then PC—OC=PO,xPO'x PO’ &c. when P is without the cirele and OC—PGC=POxPO’'x PO” x &c. when P is within the: circle. 2. Let the circumference be divided into 2 equal parts OS, SO’, O'S, &c. then PC + OC=PSXPSX &e. DEMONSTRATION. 1. Ler OC be unity, PC=«; a, a’, a’, &c. the cofines of o, OO’, OO", &c. Then will PO? = «7 +1— 24x PO?*= + 1—2ds6 &c. &c. or PO?.X PO® x &e. = x* + I—2aKX% e+ I—2a xX&e. = 4 a 7 t aa a n—tI Yo i— it 2%. x*+ 1) + ao a sate ak els - ds oe &c. n oe + 2ada" &. xx. Now if c be the cofine of any arc, the cofine of 7 times that arc a—in 1-3 n—2 nn. 2— n—5 1i—4 willbe 2 ¢c—#.2. ¢ + —2 ¢c— &c. ‘continued by (Cae fucceflively diminifhing the index of ¢ by 2 until it becomes o or I,! and - "sy ] m—u and affixing to ¢ the coefficient —— u uz u + 2.04———I.n———2% &e.to—terms avr y — 2 2 2 2,.+ when —iseveh and: u 2 Flea cna: eNO Sa 3 2 — when odd. Hence becanfe unity is the cofine of 0, P (Peri 54 i u—.in l—2 1—z2 phery), 2 P, 3 P, &c. it follows that if 2 c—z.2 c + &e. = 1 the different values of ¢ will be a, a’, a’, &c. the cofines of 0, P : : s ae &c. or that the roots of the equation I pone ie eee ae 4 a m1 =O will be a, a’, a’, &e. 2% eh24) 2+ : 2 Therefore by the nature of equations a4+a+¢@é+&c.=0 32° aa + aa +-&e. = oes Ode had A. oC, =o 4. 1—3 ‘ow Ww add a’ +&c.. a reo &e. &ew. or generally the fum of the products of zw values a,.d, a; &e. iu u “uz ¢ 2. 21—= —- I, 2 —— 2. = (to re) terms u being even = + i eaian zZ “B I id tha Ser 2. 3 ah fe wu 2 Eege ] uu = is odd and — when even: alfo the product of all the values I when 2 is odd =—>—> and when even = + 2 2 4 n ipsa) rags ee (to > terms) ; 7/4 n a—I I... Bigs" ine - — 2 2 2; Whence the value of PO? x PO” X &c. above found becomes ee 1—2 71. 1—3 n—4. n we 4T —— ax. wet y Eee eae xX + 1? @ Fie or expanding thefe terms a 2n 2n—2 her wan 2n—6 Pexere a +nx ob ileal cha aca TB eS } | I 1. 3 - - 4+2x* +1 n—2 2n—2 2n—4 Se | = nae @ ED Se dg © lg ee ee eee | VA | 22—6 l Dik 2 \ ¥ Sg te ahead ae n.u— ae yng 88 Pn, yee, 206 ty EES SS: Phe i” eet ai SEP yeh) Bee | pa, Tou Te 2, | So Ae a—6 Rape Poe ee |b ee OS Ns CoN any ei 5 ie ah Qe fo Tee ope aed re Hed J * Mr. Simpfon in his Effays, page 115, has arrived by a different procefs at a fimilar conclufion, and aflerts without any demonftration that the co-efficients deftroy each other. ‘This however is the only difficult ftep in the whole propofition, LH 35... Hence collecting the co-efficients it readily appears by con- fidering the general value of the fum of w produéts ftated above, a 2n—2m that the coefficient of the term x the fame as the coeff. of. 2m the term x — ee ae ee ee Te ie 3 = = + os m™ | 1. n—2 u—3 - = - nt—m | Se a | ra n—m- n—m+ - - 2m | TT as oon aa | N—M+1. W—M+2 - n—ramz1'! eee rem Ne me ah or reducing thefe fractions to a common denominator the numes. rator becomes # X into the expreffion in the Lemma which there- fore = o hence 2n PO: X PO? x &. =x—2x~+1 or POXxPO'x &c, EN a re Q, E. D. s 2n 2. BECAUSE PO x PS x PO’ x PS’x &. =xm1 and POx. a , 2 PO x &c.=x«m1..PSXPS’x &. = x+1. QED. bts bro. J ADDITIONAL OBSERVATIONS on the PROPORTION of - REAL ACID iz the THREE ANTIENT KNOWN MINERAL ACIDS, and onthe INGREDIENTS 7 various NEUTRAL SALTS and other COMPOUNDS. By RICHARD KIRWAN, £/. LL.D. F.R.S. and M.R.LA. al HE fundamental experiments on which the proportion of real Read :6th acid in the three mineral acids antiently known, and alfo the Te proprortion of ingredients in many neutral- falts, were determined, _ Ihave already fet forth in a paper to be found in the [Vth Vol. of the Tranfactions of this Academy. In that paper I have inferted tables of the quantity of ftandard acid exifting in 100 parts of ' each of the acid liquors, of given fpecific gravities, and alfo in each of the neutral falts therein mentioned; the mode of ex- preffing the quantity of acid I had then adopted I fince difcovered to be very inconvenient, as in fome of thefe neutral falts an acid “till Rronger than the affumed ftandard was found to exift. But A I have there alfo noticed that the ftrongeft vitriolic acid now known, exifted in viétriolated tartarin, the ftrongeft nitrous acid in nitrated foda, and the ftrongeft muriatic acid in murdated tartarin , Vor. VII. jn ne beige Acids E,2r64 ] Acids of fuch ftrength I have therefore denominated real, as either containing no water or contaizing only as much as is neceffary to their effential compofition, as far as thisis at prefent known. The method of transforming the expreflion /fandard into that of real, I have there alfo given p. 67, and by it have formed the table I here prefent ; this latter expreflion I therefore now employ in every cafe inftead of that of /fandard, together with the fubftitution of a more commodious expreffion of the ftrength of acids: The defign of this paper is alfo to exhibit an illuftration or amendment of feveral of the determinations contained in my laft, which being for the moft part fingle, required confirmation by fhewing their agreement with the experiments of feveral of the moft eminent chymifts made fince the publication of mine, that is fince the year 1791, with afew made nearly at the fame time. In my former paper I compared my refults with thofe of Bergman and Wenzel, they being almoft the only perfons who had made this fubje@ the principal object of their enquiry, and had purfued it to a con- fiderable extent; in each particular inftance I have traced the reafon of the difference of their refults from my own when it was fuch as to deferve notice, and I fhall not here repeat what I have there faid ; but I cannot avoid again mentioning one general fource _ of error attending the mode of inveftigation adopted by both and yet noticed by neither, namely the lofs that many neutral falts undergo dering evaporation, 4 lofs whofe difcovery is of con- fiderable importance, not only to the prefent inquiry, but alfo to the Limaes 4 the conduét of feveral manufactures, particularly to that of faltpetre, and hence noticed by Mr. Lavofier, 15 An. Chy. 254. On this head however I hope the Academy will foon receive the fulleft information, as our worthy member, Mr. Higgins, has at my requeft undertaken to examine its reality and extent with refpe& toa confiderable number of the moft known among thefe falts. Tuoucu Bergman and Wenzel fhould have conducted their experiments nearly in the fame manner, as far as we can judge from the mode prefcribed by Mr. Bergman in his notes on Scheffer, publifhed in 1779, yet his refults differ confiderably in many inftances from thofe of Wenzel, and appear to me far more faulty, the caufe of which feems to me to be, that he has in moft ' cafes departed from the method he had originally propofed to follow, and fuppofed quantities of water of cryftallization to exift in various fubftances without fufficient reafon, or at leaft without afligning any fuch. Thus he tells us that pellucid calcareous {pars lofe only 34 per cent. of fixed air by folution in acids, whereas the daily experience of all chymifts fhews them to lofe from 43 to 44 per cent. but 11 of thefe he fuppofes to be water, becaufe by diftil- lation he could not obtain more than 34 per cent. of fixed air, a method now well known to be defedive, as from the porofity of earthen retorts, the inefficacy of lutes, and the infufficiency of the heat applicable to thofe of glafs, the true quantity of fixed air can ne- ver be thus obtained. Mr. Cavendith could obtain from 311 grains X 2 of [ 166 J of Carrara marble only 1 grain of water*, and Florian de Belle- vue, who lately has particularly enquired into this matter, fays, marbles contain no water, or {carce any; and it is of the granularly cryftallized that he fpeakst. Dr. Watfon alfo makes the fame remark. To tartar vitriolate Bergman has alfo affigned 8 grains of water of eryftallization, whereas when dried even in a heat of 70 degrees only, except it contains an excefs of acid, it retains not even 1 percent. of water. To nitre he affigns even 18 per cent. a quantity fo great that he can-fcarce be fuppofed to have meant water of cryftallization. Lavofier, who by profeffion muft have been well acquainted with a property fo obvious, tells us on the contrary that it contains little or none, 15 An. Chy. 256. Mr. Keir allows it when not well dried about 2,5 percent. Wenzel, on the other hand, took but little notice of the water of cryftallization, and his miftakes are not fo confiderable, moft of them independently of the fource of error al- ready mentioned originated from the fuppofition of a fiGtitious fub- ftance which he called Cau/ficum, the unheeded decompofition of nitre when ftrongly ignited, and the fuppofition that acids, when the compounds into which they enter are heated to rednefs, either retain no water or at leaft a conftant and not a variable quantity of it; this is indeed an error inherent in the method purfued by him, Bergman * Phil, Trani, 1766, p. 167- $41 Roz. 94. bine? Bergman and myfelf in my firft effays. But he alfo followed ano- ther method, which preferved him from many miftakes, which was to eftimate the quantity of the ftrongeft acid in a given quan- tity of vitriolic acid, v: z: 240 grains by the quantity of it re-— tained during ignition in tartar of vitriolate, and in 240 grains mu- riatic acid by the quantity retained in muriated tartarin, for in ef- ' fe& thefe acids, as I have found, contain leaft water in thefe com- pounds ; this advantage however he fometimes loft by the decom- pofitions arifing from ignition, particularly in his experiments on metallic fubftances.- To render this paper ftill more ufeful, I fhall lay before the Aca- demy fome important determinations of the proportion of ingre- dients in compounds of which I had not myfelf treated, and are either not generally known, or fcattered in divers treatifes not-eafi- ly collected, to moft of which however I have added my own ex- periments. Wuew alkalies or earths combined with fixed air are diffolved in ' acids, though far the greater part of the fixed air is expelled dur- | ing the folution, yet fome portion of it is often retained, and may _ — in fome degree alter the fp. grav. of the folution ; this circumftance B 9 _ I did not recolleé till lately ; ; it was firft noticed by Mr. Cavendith, ‘Phil. Tranf. 1766, p- 172, and afterwards by Bergman in his notes on ‘Scheffer, §. 51, but more exp ae by ye i Chy. An. 14786, Birch Iino hae P: 13> [pene | p- 13, and by Butini on Magnefia, p. 149. As to the ufe refulting from refearches of this nature it were fuperfluous to attempt to prove it at this day, the recourfe which the moft eminent analyfts have been obliged to have to it in particular inftances, as will pre- fently appear, fufficiently evinces it. ‘ Inquiries of this kind (fays “© Mr. Fourcroy) are more difficult and delicate than thofe which “© have hitherto been made on falts; whatever requires a precife “ knowledge of quantities and proportions, prefents difficulties fo “ great as often to appear infurmountable, yet without this know- “ ledge no progrefs can now be made in chymiftry,” 10 An. Chy. 325; and according to Bergman, “ Ufus cognite proportionis prin- “ cipiorum ingredientium egregius eft et multifarius.” 1 Bergm. 137. chap. 1. § 1. TABLE be ; ay a ! * ee ; ; A antl : - s , 7 t i Paes _ ; « . , = ‘ ‘ > cal ~ . . ‘ fo ! , . ae a aT o> pa at Mp veg SFL ST > i ae pete Seat gee xy [To face Page 168. 1G A B L E OF THE QUANTITY OF REAL ACID In 100 Parts of Vitriolic, Nitrous and Marine Acid Liquors of different Denfities, at the Temperature of 60°. | ae 5 In Fitriolic Acid of different Denfities, at || In Nitrous Acid of different Denfities, at the or Ae: of ae the Temperature of 60°. Temperature of 60°. Tenaertave of 608. 100 Parts | Real 100 Parts | Real 100 Parts Real too Parts Real roo Parts Real Sp. Grayity.| Acid. |Sp. Gravity.) Acid. |/Sp. Gravity.) Acid. |Sp. Gravity.| Acid. Sp. Gravity. | Acid. ——— -—|— —e 2,0000 |89,29 1,4666 144,64 155543 73954 153364 41,91 1,196 25,28 1,9859 |88,339 154427 143,75 1,5295 69,36 1,3315 41,18 1,191 24,76 149719 |87,50 1,4189 |42,86+ 1,5183 69,12 1,3264 40,44. 1,187 24,25 1,9579 86,61 1,4099 |41,96 1,5070 68,39 1,3212 439,71 1,183 23573 1,9439 |85,71 I,4010 |41,07 1,4957 67,05 1,3160 38397 ‘1,179 23,22 1,9299 184,82 1,3875 40,18 1,4844 66,92 1,3108 38,34 1,175 22,70 1,9168 |83,93 1,3768* |39,28 14731 66,18 1,3056 375° 1,171 22,18 1,9041 |83,04+]| 1,3663 138,39 14719 65545 1,3004 36577 1,167 21,67 1,8914 52,14 153586 137,50 1,4707 64,71 1,2911 36,03 1,163 21,15 1,8787 {81,25 | 153473 |36,60 1,4695 63,98+ | 12812 35,30+|| 15159 20,64 1,8660 80,36 1,3360 |35,71 1,4683 63524 1,2795 34350 1,155 20,12 1.8542 79546 1,3254 34,82 1,4671 62,51 1,2779 33,82 1,151 19,60 1,8424 {78,57 1,3149 133393 1,4640 61,77 1,2687 33509 | 1,147 19,09 1,8306 |77,68 1,3102 |33,03 1,4611 61,03 152586 32.35 1,1414 18,57 1,8188 |76,791+]) 1,3056 32,14 1,4582 60,30 1,2500 31,62 1,1396 18,06 1,8070 |75,89 | 1,2951 131,25 154553 59556 | 1,2464 30,88 1,1358 17554 157959 |75>— 1,2847 30,35 154524 58,83 1,2419 30,15 || 151320 17302 1,7849 74511 1,2757 29,46 144471 58,09 152374 29,41 151282 16,51 157738 |73,22 1,2668 —|28,57 1,4422 57530 31,2291 28,68 1,1244 15,99 1,7629 |72,32 1,2589 |27,68+ |) 1.4373 56,62 1,2209 -| 27,94 1,1206 15,48 1,7519 {71,43 1,2510 26,78 1,4324 55589 1,2180 27,21+ 1,1168 14,96 1,74160 |70,54+] 152415 |25,89 1,4275 55515 1,2152 26,47 LE prwss) 14544 1,7312 |69,64 11,2320 |25,— 1,4222 5412+] 1,2033 25,74¢ 1,1078 13,93 1,7208 168,75 31,2210 |24510 1,4171 53,68 1,2015 25,00 1,1036 13,42 1,7104 67,86 1,2101 |23,21 1,4120 52594 1,1963 24,26 1,0984 12,90 1,7000 66,96 1,2009 |22,32 1,4069 52,21 I,1911 23,53 1,0942 12,38 1,6899 |66,07 1,1918 2143+ \| 154018 51547 1,1845 22,79 1,0919 11,86 1,6800 65,18 1,1836 — |20,53 1,3975 50574 1,1779 22,06 1,0868 11935 1,6701 |64,28 1,1746 19,64 1,3925 50,00 151704 21,32 1,0826 10,83 31,6602 63,39 1,1678 {18,75 1,3875 49,27 1,1639 20559 1,0784 10,32 1,6503 |62,50 1,1614 17,85 1,3825 48,53 1,1581 19,85 1,0742 one 1,6407 61,61 1,153t {16,96 1,3775 47,80 1,1524 19,12 1,0630 oe 1,6312 |60,71 1,1398 |16,07 1,3721 47,06 1,1421 18,48 1,0345 Sot 31,6217 {59,82 1,1309 15,18-+ 1,3671 46,33 1,1319 17,657 1,0169 2,58 1,6122 = |58,93 1,1208 |14,28 1,3621 45559 1,1284 16,91 a ee Lain 1,6027 158,03 1,1129 13,39 1,3571 4486+] 1,1241 16,17 31,5932 |57514 1,101 |12,50 1,352 44s12 1,1105 15544 1,5840 |56,25 1,0055 11,60 __ 1,3468 43338 1,1111 14,70 15748 |55,36-+| 1,0896 |10,71 1,3417 4265 1,1040 13527 1,5656 |54,46 1,0833 | 9,80 eel 15504 |53.57 1,0780 8,931 || The Numbers above the Lines drawn acrofs 13,5473 |52,68 1,0725 8,03 the Tables of vitriolic and nitrous Acids were! 1,5385 151,78 1,0066 7214 |\found by Experiments; thofe under the Lines 1,5292 50,89 1,0610 | 6,25 |lonly by Analogy, ¥,5202 50,00 1,0555 5935 155112 |49,11+4+] 1,0492 4:46 ae ba esa ae The Affinity of Vitriolic Acid to Water de-' 14844 45,43 1,0343 147 creafes in the Ratio of the Square of the Quan-| 144755 145,53 tity of Water united to it. 23 Ann. Chy. 196 aul 299) - — |jand 197. Pi A . And fol believe it does to all other Subftances ; pate rae CRSA SED A ES RG cae Affinity that is commonly given, Note. The ftandard Quantities of Vitriolic Acid were reduced to Real by multiplying them int 8929. of the Nitrous, by multiplying them into 0,7354, and the Marine by multiplying them into tee far ue Realone raeationed in my laft Paper. Y—1 7 py 265°] Of ithe Alteration arifing from Difference of Temperature. To difcover this alteration by experiment in each individual in- ftance would be an endlefs tafk, hence I have felefted only 3 cafes with refpect to the vitriolic acid, and 2 of the nitrous, and obferved. the changes in each at every 5 degrees above 60° unto temperature 70°, and at every 5 degrees below 60° unto temperature 50° nearly, thefe being the temperatures at which experiments are ufually made. Of the Vitriohe Acid. Vitriolic acid ‘ 1,8360 at temperature 60° Becomes 1,8292 at - 70° ' 1,8317 at - 65° 1,8382 at - bg 1,8403 at - 50° 1,8403 at - 498 hence we fee that vitriolic acid, whofe denfity at 60° is 1,8360, lofes by a/cenaing and gains by de/cending 0,00068 for every degree of temperature between 60° and 70% and 0,00043 nearly by each ‘degree between 609 and 49°. Vou. VII. me Again [hyo Again, vitriolic acid 1,7005 Becomes 1,6969 1,6983 157937 1,7062 at 60° 70° 65° 55° 50° hence vitriolic acid, which at 60% is 1,7005 gains or lofes 0,00036 neatly for every degree between 60° and 70°, and 0,g0051 by every degree between 60° and 50°. Laftly, vitriolic acid 1,3888 Becomes 1,345 1,3866 1,3898 1,3926 at at at at at 60° 70° 652 55° 49° hence vitriolic acid, which at 60° is 1,3888 gains or lofes 0,00043 nearly by every degree between 60° and 70°, and 0,00034, nearly by every degree between 49° and 60°, between 49° and 50°,'I per- ceived no difference. Of the Alteration of Denfity from Difference of Temperature in — Nitrous Acid. Nitrous acid, which was 1,4279 Became - - 1,4178 14225 ~ : at at at 602 70° 65° a ae ee [ x7 | £,4304 at - 55° 1,4336 at - 509 1.4357 at F 45° hence nitrous acid, which at 60° is 1,4279, gains or lofes 0,oo101 neatly by every degree between 60° and 70°; and 0,00052 by every degree between 45° and 60°. I formerly found that the ftrongeft /Pirit of nitre ig moft ex- panded by heat or contra€ted by cold. Atso, that nitrous acid, whofe fp. grav. at 34° was 1,4/750, was expanded by heat as follows : 1.4750 at an then it gains or lofes 0,0097 by 15° became 1,4653 at 49 between 34° and 49° inclufively. Again I found that colourlefs nitrous acid whofe {p. grav. was 1,4650 at - 30° became - 1,4587 at - 46° 1,4302 at - 86° hence by the firft 16° from 30% to 46° it gained 0,0063, and by _ 40°, that is from 46 to 86°, it gained 0,0285. Again, nitrous acid whofe denfity was 1,2363 at. - 60° became’ = -'1,2320* af... - 70° 1,2342 at = 65° Yaw | : » 71,2384 come | 1,2384°> at = 55° 1,2406 at - 50° 1,2417 at - 45° hence nitrous acid, which at 60% is 1,2363, gains or lofes by every degree between 60° and 70°, 0,00043 and 0,00036 by every degree between 60° and 45°; and we may affume 0,0005 as the variation incident to every degree between 60° and 70° in nitrous acid, whofe denfity at 60% is between 1,3 and 1,4 and 0,0004 for the variation between 44° and 60% Of Marine Acid. I formerly found that this acid of the denfity 1,196 at 33° became of the denfity 1,1820 at 66°, the alterations of acids of lower fp. grav. I have not examined, but I found that in general its dilatability is greater than that of nitrousacid of the fame denfity. OF THES USE OF THESE) TABLES Prosiem tit. An extratabular {pecific gravity being given, but intermediate between fome of thofe in the table, to find oe qeantity of real acid in 100 parts of fuch acid liquor. rft. Rays J 1ft. Finn the difference betwixt the next higher and lower tabular denfities = D, and alfo the difference betwixt their acid contents = D’. ad. Finp the difference betwixt the extratabular fp. gravity and the next upper or next lower, which ever it is neareft to = d, and let the difference betwixt its acid contents (or quantity of real acid) and thofe of the next upper or lower = a’, which is the , quantity fought ; then as D. D’:: @. a’ then d= = 4 confequent- ly a’ added to the acid contents of the lower tabular fp. grav. or fub- ftracted from the upper, zs the quantity fought. Note. In general when d, that is the difference between the ex- tratabular fp. gray. and any tabular fp. grav. does not exceed +255 it is infenfible, and the acid contents of the lower or upper, which _ ever is neareft, may be afcribed to it. ProsLiem 2d. THE quantity of real acid in 100 parts of an acid liquor being. given but extratabular, being intermediate between fome of the quantities in the tables, to jind the fp. grav. of fuch acid liquor. Dd mo. a, d added to the ater tabular fp. grav. or fubftracted from the upper, gives the fp. grav. fought. . © Fiyp D, D’ and d as‘ in the cae problem. then d= Bur [ 174 ] Burt with regard to the marine acid its fp. grav. is to be invefti- gated according to the ordinary mathematical rules. PROBLEM 3d. To find how much water muft be added to too parts of an acid liquor of a given fp. grav. to bring it down to another lower given {p. grav. ft. Frnp by the table the quantities of acid and water in 100 parts of each of the acid liquors refpectively, each being fuppofed to be in the table, let the quantity of water in the denfer be W, and the quantity of acid = A, let the quantity of water in the lefs denfe = w, and the quantity of acid =a, and the quantity of water to be added to 100 parts of the denfer = m then W + m mutt be to Aas w toa And Watam=Aw. Andam=Aw—Wa. Aw—Wa And m= a ProgLeM 4th. Given weights of 2 or more acid liquors of different fp. gravities being mixed, to find the quantity of real acid in 100 parts of the mixt liquor and its fp. grav. FInp [i756] Finp the fum of the quantities of real acid in 100 parts of the mixture, then find the refulting fp. grav. by the 2d problem, if the given fp. gravities be extratabular, the operation muft be more te- dious, as the acid contents of each muft be found. Prosiem Sth. THE quantity of an acid liquor requifite to faturate 100 parts of aay bafis being found, to find the fp. grav. of that acid liquor. 1ft. Finp by the 4th table the quantity of real acid requifite to faturate 100 parts of the given bafis, it is then plain that the given quantity of acid liquor contains the requifite quantity of real acid, fince it is fuppofed to faturate 100 parts of the bafis and hence we may fee how much too parts of fuch acid liquor contains of real acid, and if this laft found quantity be in the table, its fp. grav. will be feen, but if extratabular, its fp. grav. muft be fought by the 2d problem. ProsyieM 6th. THE quantity of real acid requifite to faturate too parts of any bafis being known, to find how much of one acid liquor of any given fp. grav. is requifite to faturate that, and confequently any other given quantity of fuch bafis. If Lamen6: | If the given fp. grav. of the acid liquor be ¢abular the quantity of real acid in 100 parts of it is apparent, and confequently the quantity of fuch acid liquor containing the required quantity of real acid, is eafily found by the rule of proportion. But if the given fp. grav. is extratabular the quantity of real acid in 100 parts of the acid liquor muft be fought by the firft problem. ProgiemM 7th. Tue quantity of real acid, in a given quantity of an acid liquor being known, and alfo the quantity requifite to faturate 100 parts of any given bafis. To difcover the quantity of fuch bafis con- tained in any folution, or in any powder, by which the given quantity of acid liquor is faturated. Ir the bafis be fingle (that is unmixed with any other bafis to which the acid may unite) or combined only with fixed air the folution is eafy, but if the given bafes be mixed with other bafes combinable with the fame acid, the folutien is more complex and varies according to the variety of cafes. PRoBLeEM 8th. To find how much of an acid liquor of one fort will hold as much real acid, as is held by a given weight of an acid liquor of another Jort whofe fp. grav. is alfo given :—For inftance, how much vitriolic acid will contain the fame quantity of real acid as is contained in 100 grains nitric acid whofe fp. grav. is 1,3925. . ait Loagy od rt. Firft find by the table the quantity of real acid contained in. the given quantity of the fecond acid, whofe fp. gr. is given, or if not in the table it muft be found by Problem rft. ad. It is apparent that the quantity of the firft acid liquor muft vary with its fp. gr. thus, in the inftance given, as 100 parts nitrous acid of the fp. grav. 1392 contains 50 parts real. nitrous acid, fo 100 parts vitriolic acid whofe fp. grav. is 1,5202.contains by the table. the fame quantity of real acid, v. z. 50 parts, but of the vitriolic acid whofe fp. grav. is 1,800 only 64 parts are. requifite to contain 50 parts of real acid, whereas 200 grains are requifite of the vi- triolic acid whofe fp. grav. is 1,2320. Note, The folution of this problem may hereafter be found of ufe in comparing the quantities and. affinities of oxygen in different acids. ProsLeM gth. To find the fp. grav. of fuch vitriolic acid as that too parts of - it fhall contain the fame quantity of real acid as 100 parts of the nitrous. Tuts. can be found only by in/peétion on confulting the ta- bles; an example has been feen in the laft problem, fo alfo. 100 parts vitriolic acid 1,3102 contain the fame quantity of real. acid as 100 parts nitrous ,acid whofe fp. gr. is 1,2687.. And. 1oo grains vitriolic acid whofe fp, gr. is 1,1746. contains the. Vou. VII. ee Me fame- teh] fame quantity of real acid as 100 grains {p. falt whofe fp. gr. is 1,159. AnD 100 grains nitrous acid 1,1963 contains the fame quantity of real acid as 100 grains fpirit of falt whofe fp. grav. is 1,187. Hence it fhould feem that the fp. grav. of the real marine acid is {maller than that of the real nitrous, and that of the real nitrous fmaller than that of the real vitriolic, fince when the weight of each acid, and alfo the weight of real acid in each is equal, the vi- triolic acid is fpecifically heavier than the nitrous, and the nitrous than the marine, but this perhaps may arife from penetration. : PRoBLEM I0. To find how much of a neutral falt of one fort holds as much real acid or bafis as a given weight of the /ame neutral falt in another flate, or as a given weight of another falt in any given ftate. Tuese queftions are refolved by the 4th and sth tables, thus if it be afked, how much nitre contains as much acid as 20 grains of vitriolated tartarin? By the 4th table I fee that 221,48 parts of vitriolated tartarin and 227,22 parts nitre contain equal quantities of acid fince both contain *100 parts, then as 221,48, 227,22:: 20.20,5. Ms ria Le AGAIN, ‘E79 J Acatn, How much deficcated foda will hold as much alkali as 30 parts cryftallized foda? Inthe 5th table I fee that 541,1 parts of ‘the cryftallized hold as much alkali as 227.4 parts of the defic- cated, then as 541,1 . 227,4:: 30. 12,6. ProsureM IIth. How much of a given bafis will be requifite to faturate the acid contained in a given quantity of a given neutral falt, thus how much deficcated foda will be requifite to faturate the acid contained in 50 parts cryftallized Epfom? By the 4th table I fee that 100 parts real vitriolic acid are con- tained in 340 parts cryftallized Epfom. Then if 34c.100: : 50, 14,7, then by the 3d table I fee that 100 grains of real vitriolic acid faturate 78,32 of foda, Confequently if 100 faturate 78,32 :: 14,7 would faturate 11,51 of foda. Laftly, In the 6th table I find that roo grains deficcated foda - contains 60 of foda. Thenif 100: 60::%. 11,51, then «= 19,1 parts deficcated foda. Then 19,1 parts deficcated foda will faturate the acid contained in 50 parts cryftallized Epfem. Note 1ft. This problem is of ufe in determining the quantity . of any Sheen” fubftances to be employed in decompofitions, — Z 2 operated. f° 280° | operated either by a fingle or double affinity. But in moft cafes more of the precipitant muft be employed than the exa quantity neceffary for faturation, and particularly when decompofitions are attempted in the dry way, as otherwife a complete contac with the fubftance to be decompofed will not be attained, or if volatile it may be fublimed before the decompofition takes place. Prosiem 12th. Some analyfts have denoted the ftrength of their acids by ex- prefling the quantities of each neceffary to faturate a certain quantity of alkaline liquor (and fometimes of another bafis) without even telling whether the alkali was mild or cauftic, or the quantity of it contained in the alkaline liquor. This problem is confequently indeterminate. However a method of giving fome folutions of it may be underftood from the following example ; and circumftances will generally fhew whether the application to particular cafes be juft. Link tells us that 240 grains of a vitriolic acid which he employed, ' faturated 6,5 times its weight of tartarin (he muft mean in a liquid ftate, as no vitriolic acid will faturate fix times its weight of real alkali) and that 240 grains of the nitrous acid he employed fatu- rated 2,5 times its weight of the fame alkali. Quere’ the fpec. gravity of both acids? 1ft Ir is plain, that fince 240 grs. of the nitrous‘acid faturated 2,5 times its weight of the alkali, 624 grs. of that acid would fa- turate ae So Page J _turate 6,5 times its weight of the alkali; and fince 624 grs. of the nitrous acid would faturate as much alkali as 240 of the vitriolic. acid, then 260 grs. of it would faturate as much alkali as 100 gts. of the vitriolic acid could faturate. Therefore fuppofing 100 of the vitriolic acid to contain 75 of real acid, fince more real nitrous acid is required to faturate a given quantity of tarta- rin than of vitriolic acid, in the inverfe ratio of 1214 to 1 a (as appears by the third table,) then denoting the quantity of real nitrous acid in 260 grs. of the nitrous liquor, by x we have the following equation as 1214. 1177 :: * 75. and p= 377,55. Then 260 grs. of the nitrous acid contain TSS of real nitrous acid, confequently roo grs. of it contained 29,82 real acid. And therefore its fp. grav. was nearly 1,234, and that of The quantity of alkali in the alkaline the vitriolrc about 1,800. liquor might alfo on this fuppofition be determined. So if it be required to know how much common {alt is requifite to decompofe a folution of nitrated filver containing 176,25 grs. of filver: q tft. I find by the 6th table that 75 grs. filver take 16,54 of ma- rine acid, confequently 176,25 gr. filver take up 38,87. ad. By the 4th table, I find that too grs. muriatic acid are con- tained in 257,2 of common falt, confequently 38,87 are contained in 99,973;,. that is 100 grs. common falt, then roo grs. of it are neceflary to precipitate the filver. ae ILLUSTRATION [ 182 ] ILLUSTRATION OF THE TABLES Few chymifts have made experiments appofite to my prefent purpofe, and thofe that have made any relative to it, have gene- tally neglefted marking the temperature, and thus prevented an exact comparifon of the refults they obtained with thofe that fhould be expected from the quantities of real acid and water fet forth in my tables. Tue moft accurate of thefe experimenters was Hahn, who has inftituted a confiderable number, of which an account is given in his Differtation De Eficacia Mixtionis in Mutandis Corporum Voluminibus, of thefe I fhall fele@ a few, which I think by their coincidence with the refults to be obtained, calculating from my tables, furnifh a full proof of their accuracy, at leaft to as great a degree as can be expected in fubjects of this nature. Or THE TaBLE or ViTRIOLIC AcrD. iff Experiment. Haun, to 800 grs. of vitriolic acid whofe fp. grav? was 1,848 at the temperature of 44°, added 400 grs. of water in a veffel that confined the vapours, and when the mixture was cooled down to the temperature of the air he found its fp. grav. 1,545.—p. 48 and 49. ; Application. bs J Application. Virriouic acid of this denfity lofes, as we have feen, ,00043 in denfity, by each degree between 44° and 60°; hence its fp. grav. at 60° fhould be 1,8489 —,00043 X 16 = 0,0068 = 1,8421, which differs infenfibly from the next lower tabular fp. grav. 1,8424, and therefore this may be taken for it. Tue quantity of real acid in 100 grs. of the acid liquor, whofe fp. grav. is 1,8424 amounts to 78,57 per cent. per table, then 800 gts. of that acid liquor contains 78,57 x 8 = 628,56 of real acid, and confequently the 1200 gr. of the mixture contain that quan- tity of real acid, and therefore 100 grs. of the mixed liquor con- tain 52,38 of real acid, which we fee differs but little from the tabular real acid, 52,68 which indicates the fp. grav. to be 1,5473, and the difference between this and the fp. grav. found by Hahn is inconfiderable. | However, to obtain a clofer approximation, and to give an ex ample of the made of folving the 2d problem, I fhall deduce the fp. grav. from the rules laid down for the folution of that problem. 1ft. Tue next higher fp. grav. is 15473, and the next lower is 1538s, and the difference between them is 0,0088 = D, Their acid contents are 52,68 and 51,78, and their difference 0,9 = D. 2d. [ 184° | ad. Tue difference betwixt the given extratabular acid contents, 52,38, and the next lower tabular acid contents 51,78 is 0,6=@ then @, the quantity to be added to the lower fp. grav. is found by the formula d= a he 0,0088 x 0,6 _ 0,0528 = 0,0058 059 0,9 17,5385 Now + 0,0058 and that found by Hahn is 1,545. = 1,5443 Tuis it is true would be the fp. grav. at 60°, and after 3 days reft (the time I allowed for the penetration of the mixtures men- | tioned in my tables,) and it does not appear what the temperature of Hahn’s mixture was when he took its fp. grav. if it was 44° (the temperature of his oil of vitriol) it is poffible that the cold without exact penetration might produce an effect equivalent to that which time would produce by penetration, ; 2d Experiment. In this Hahn added 400 grs. of water to the 1200 grs. of the foregoing mixture, and confequently the new mixture weighed 1600 grs. and contained the fame quantity of real acid as the fore- going, that is 628,56 grs. he found its fp. grav. when cold to be 138,40. Application. f an Application. Since 1600 grs. of the mixture contained 628,56 real acid, 100 grs. of it fhould contain 39,28 ; now this quantity of real acid is exadtly in the table, and correfponds with the fp. grav. 1,3768. Then the difference between Hahn’s refult and that of my deter- 3) NE st eRe i minations is —_—. 19000 3d Experiment. To the 1600 grs. of the laft mixture Hahn added 800 of water, and when the whole was cooled down to the temperature of the air he found the fp. grav. of the mixture 1,2439. Ibid. p. so. Application. Turis mixture weighed 2400 grs. and contained the fame quan- tity of real acid as the laft, namely, 628,56 grs. confequently 100 grs. of it contained 26,19; this quantity of real acid is extratabu- lar ; the neareft tabular quantity of real acid is 25,89, which cor- refponds with the fp. grav. 1,2415; though this feems fufficiently near to Hahn’s refult, yet I have found it more exa@tly by the ad problem. Here D = 0,0095 and D’= 0,89 and @’= 0,3,'then by the furmula d = — we have SS eos oe 0,0032, and the lower fp. grav. 1,2415 +0,0032 = 1,2448, which differs from Hahn’s refult by only OTe ae! ; 10000 Vou. VIL. Aa , Tis P2186 |] Tue 3 firft experiments of Hahn not perfe@tly agreeing with each other, and not having been made with equal accuracy, I omit. Morveau’s Experiment on the Quantity of Real Acid in Vitriolie Acid, whofe Sp. Grav. was 1,841. 1 Encyclop. 592. ~ He took 58 grs. vitriolic acid, whofe fp. gray. at 8°,5 Reaum. (= 51° Fahr.) was 1,841, and poured into it a folution of acetited barytes until a precipitate ceafed to appear. The precipitate wafhed and dried (by ignition as it would feem by what he adds in the 2d column of the above page) weighed 110,3 grs. Application. Vitriotic acid, whofe fp. grav. at 51° of Fahr. is 1,841, would have its {p. grav. lowered to 1,838 at 60° of Fahr. the degree for which my tables were formed, as I have fhewn in my remarks on the alteration by temperature. Now the fp. grav. 1,838 is intermediate between the tabular denfities 1,8306 and 1,8424, but nearer to this; then by the firft problem its acid contents will be found to be 78,24 per cent. then if. 100 grs. of, vitriolic acid of this fp. grav. contain 78,24 per cent. real acid, 58 fhould contain 45,37 of real acid. But 110,3 grs. of ignited barytes contain 36,76 real acid, allowing 100 ers. of fuch barytes to contain 33,33 per cent. the difference then be- tween Bee a tee et ee Rae f Come | '. tween Mr. Morveau’s refult and that of my calculation is 8,61 grs.; the reafon, however, is*obvious; Morveau employed acetiid barytes, this acid rendered part of the acid fulphureous, as is well known ; the fulphureous acid does not decompofe acetited barytcs per Bergman’s table, his other experiments on the fulphureous acid cannot therefore apply. Or THE TasiLe or Nitrous Acip. Tuoucu this acid was not exa‘tly oxyginated and colourlefs, yet it was far from being fully de-oxyginated, but in that pale red ftate in which it commonly appears; what changes the variety of oxygi- nations may produce [ have not experienced ; the refults are not quite fo accurate as moft of thofe in the table of vitriolic acid, partly from the eruption of vapour during the weighing, and-partly from the diforder the fumes caufe at long run in the {cales; but the error in the quantity of real acid in 100 parts of the acid li- quor,” no where, as far as I have had occafion to examine, amounts to 1 per cent. or at leaft does not exceed that amount; the lower part of the table I found moft faulty, and have reCtified the _ errors to a great degree. Experiment ft. To 400 ers. of nitrous acid, whofe fp. grav. at 63° was 1,4995, _ Hahn added 200 of water, and when the whole was cooled down to Me he found the fp. grav. to be 1,3157- : Aa2 Application. D788, } Application. The fp. grav. 1,4995 at 63° would be (by the table of variation already feen) 1,4995 + CoI10I X 3 =1,5025, which fecarcély dif- fers from 1,5070, a tabular number, which denotes the acid con- tents 68,39 —and if 100 grs. of this acid liquor contain 68,39 real acid, 400 grs. contain 68,39 x 4 = 273,56, and when 200 grs. of water were added, then 600 grs. contained 273,56, and confe- quently 100 grs. of the mixture contained 45,59, which indicates the tabular fp. grav. 1,3621, which at the temperature of 64° would be 1,3581. Turis denfity differs much from that found by Hahn, being 42 TOOO?, but that the error proceeds from his not having allowed fufficient time for the penetration of the water and acid, and from the lofs of acid by the heat excited will be feen in the examination of the ad experiment. " Experiment 2d. To the 600 grs. of the mixture of the laft experiment, whofe {p..grav. was by him 1,3157, and at 60° would be 1,317, he added 200 prs. of water, and found the fp. grav. of this laft mixture at 64°, 1,2561, which at 60° would be 1,2578, thegheat excited amounted to 80°. Application. Peer.] Application. Tue fp. grav. 1,317 differs infenfibly from 1,316, which indi-~ eates the acidity 38,97 per cent. and if roo grains contain 38,97:: 600 fhould contain 233,82 (whereas we have already feen that 600 contains 273,56) and when 200 grains more of water were added, then 800 fhould contain 233,82, and confequently too fhould contain 29,22 real acid, which indicates very nearly the fp. grav. of this 2d mixture to be 1,237, which differs from Hahn’s refult 1,257 by 8257 2° _, a difference which, though confiderable, is 0,020 ~ 1000 by the half {maller than that of the rft experiment, as by the in- terval of time between the 1ft and 2d- experiment the penetration of the 200 grains of water firft added had increafed.. Turs calculation is grounded on Hahn’s refults, which are erro- neous from want of reft and the efcape of vapours. We fhall now _ fee what tke fp. grav.. of this laft mixture fhould be, if both this and the former experiment were more accurately conduéted, and the water fo gradually added that little or no heat weuld be gene- rated, on which principle my former calculation proceeded. ‘This ex- periment mayybe confidered as a mixture of 600 grains of an acid liquor, whofe fp. grav. fhould, by my table, be 1,3621, and whofe acid contents are 273,54 grains with 200. grains of water, and then 800: [ 190 ] Sco grains (the quantity of this 2d mixture) muft contain 273,54 grains of real acid, and confequently roo grains of this new mix- ture contains 34,19 grains real acid, which indicates very near the fp. grav., 1,2779, which differs from Hahn's refult By 22 e one fo much high r. Bor this fame experiment may alfo be confidered as a mixture of 400 grains of the ftrong acid 1,5025 with 400 grains of water, then as the 400 grains acid liquor contains 273,56 grains real acid as already faid, 800 grains of the mixture fhould contain the fame quantity of real acid, and the fame fp. grav. would be found to refult as above. Experiment 3d. Iy this experiment he added 2 parts water (fuppofe 200 grains) to 1 part of the fp. of nitre 1,5025, much heat and copious red vapours were produced, infomuch that ‘a few grains of the weight of the whole were loft (about 3 per cent.) and the fp. grav. was I,1723, the temperature is not mentioned, but it feems probable it was 64°, the temperature at which, he fays, the mixture was made, then at 60° it «would be 1,1740. Application, Here the 300 grains of mixed acid liquor ‘contained 68 539 ——— [33 i TueENn 100 grains Leucite fhould contain, per Bergman, 21,35, and by my calculation, 22,4. By Bergman’s calculation there is a deficit of 0,27 of a grain, and by mine an excefs of 0,77 of a grain. See 2 Klaproth, so. Bur with refpe@ to nitre, my calculation has the advantage both over his and Wenzel’s, for fince 300 grains Leucite afford 123 of nitre, too grains of this ftone fhould afford 41. Then by Berg- man’s account, 41 grains fhould contain 20,09 of alkali, which leaves a deficit of 2,16 grains; by my determination 41 grains of nitre contain 21,238, which leaves a deficit of only 1,012 grains: [Silex §3,50] Silex’ 53,50 For the calculation ftands thus:d Argil 24,25 bArgill 24,25 {_Tartarin 20,09 J Tartarin 21,238 97584 98,988 ————s SeGiion 2d. NitRaTED Sopa. . In my former experiment 36,05 grains of mere foda were fa- turated by 145 of nitrous acid, whofe fp. grav. at 60° was 1,2574; this. denfity is intermediate between the tabular fp. Vou. VII. Gg gravities [epee]: gravities 1,2779 and 1,2687, but nearer to the former, and by the folution of the 1ft problem will be found to denote 33,8 grains real acid; confequently 145 grains of this liquor contained 49 grains of real acid. The quantity of nitrated foda formed was found by a ftandard experiment to be 85,142 grains, which is very nearly the fum of the weights of the real acid and mere alkali, as 36,05 + 49 = 85,05; this trifling difference may be water. Hence 1oo parts deficcated nitrated foda fhould contain 57,57 real acid and 42,34 foda. THEN Ioo parts foda fhould take up 135,71 of real nitrous acid. And Ico parts nitrous acid fhould take up 73,43 of foda. SuRpRIsED at finding no water in this neutral falt, I lately examined its compofition by my antient method. I diffolved 200 grains of pure and well deficcated foda, and faturated the folu- tion with 1225 grains of dilute nitrous acid, of which 4 confifted of the concentrated acid 1.416, of which confequently 306,2 were employed; the lofs of fixed air was 75 grains, and confe- quently the quantity of real alkali was 125 grains. Tue {p..grav. 1,416 lies between the tabular gravities 1,417 and 1,412, and by the folution of the 1{t problem its centenary, quantity of acid will be found to be 53,53; and t100 grains of this L295 J this acid liquor being diluted with 300 of water, then 400 grains ~ of the dilute acid contain 53,53 realacid; and confequently 1225 grains of it contained 163,9 grains; this quantity therefore was taken up by 125 of foda; if therefore. the falt thu® formed con- tained no water the fum of both thefe quantities fhould exprefs its weight, namely, 163,9 + 125 = 288,9 grains ; but having very gently evaporated the folution, namely, in a heat not exceeding 120°, and then drying the refiduum in a heat of 400° for fix hours, I found it to weigh in the evaporating difh (from which I could not feparate it without lofs) 308 grains, confequently thefe 308 grains contained 19,1 of water, however it is evident that in a greater heat even thefe would be evaporated. Anp then very nearly the fame proportion of acid and alkali would be found as in the preceding experiment, for 308 — 19,1 = 288,9, and if 288,9 grains contain 125 of alkali, roo grains of the nitrated foda fhould contain 43,27, and confequently 56,73 of acid; and allowing 19 grains of water in 308 of this falt dried at 400°, then too grains nitrated foda fhould contain 40,58 of foda, 53,21 of real nitrous acid, and 6,21 of water. _Anp 100 grains mere foda faturated with nitrous acid fhould afford 246,42 of nitrated foda dried at that heat. And roo grains real nitrous acid faturated by foda fhould give 188 nearly of ni- trated foda dried as above. . Gega2 BERGMAN fe 3.6 ° 3 BerGMAN, Vol. I. p. 20, allows to 100 parts foda very nearly the fame quantity of real nitrous acid asI do, namely, 135,5 parts. Havine re-diffolved the above 308 grains and expofed the fo- lution to fpontaneous evaporation, I found the cryftals dried at 70° to weigh 317 grains; hence this falt contains 2,8 per cent. of water of cryftallization, but in a ftrong heat it would lofe much more. Tuoucu Wenzel’s determinations feemingly differ confiderably from the foregoing, yet on a clofer infpection the difference will be found not greater than the ufual imperfection of weights and weighing, and the varying nature of the acid may admit. He found 71,5 grains mere foda faturated with nitrous acid to afford 190,75 of thoroughly deficcated nitrated foda, and hence concluded that it contained no water, and confequently 190,75 grains of this falt to contain 71,75 of alkali and 119,25 of real acid. Hence 100 grains of this fait fhould contain 37,48 of alkali and 62,52 of acid; it is plain then that this acid contained the 6,21 grains of water which I found in 100 parts of this falt, for if we add 6,21 to the quantity of acid I afcribe to 100 parts of this falt, we fhall find very nearly Wenzel’s weight of ig sa 54,01 + 6,21 = 59,42. Accor DING ee ee ee ee ff aay. J AccorpinG to Wenzel, then, too grains mere foda take up 166,7 of this aqueous acid, and fhould afford 266 of nitrated foda thoroughly deficcated ; and 100 grains of the aqueous acid fhould take up 59,9 of foda. From: the experiment on nitrated foda Wenzel deduces the ftrength of his fp. of nitre, which being the fame as he employed in his fubfequent numerous experiments it is important to dif- cover. As he faturated 71,5 grains foda with 347 grains of this fp. of nitre and found the foda to take up 119,25 of what he thought the ftrongeft nitrous acid, he concluded that 240 grains of it con- tained 82,5 of the ftrongeft acid, and confequently 100 grains of it fhould contain 34,375 of his ftrongeft acid. Now to compare the quantity of his real acid in his fp. of nitre with that which 1 judge his to poffefs, I muft obferve that to faturate 71,5 grains mere foda, 96,933 grains of my real acid would be requifite, and confequently that 347 grains of his fpirit of nitre contained no more ; therefore 240 grains of his fpirit of nitre contained but 67,04 of my real acid, and 100 grains of it contained 28 of my real acid; the difference is water contained in his ftrongeft acid. ‘Then 1000 grains of his ftrongeft acid is only equal to 812,6 of my real acid; the remainder v. x. 187,4 being water containedvin his firongeft acid. ’ Morveav. ; iti LU geal Morveavu faturated 485 grains of cryftallized foda (= 104,66 of mere foda) with 545 grains of fpirit of nitre, whofe fp. grav. at 4° of Reaumur (= 41° of Fahrenheit) was 1,2247, which at 60° of Fahren. would be 1,21g, and this by my calculation contains about 27,29 per cent. real acid; confequently 545 grains contained 148,73; then roo grains mere foda fhould take up 142 of real nitrous acid. 2d Old Mem. Dijon 184. Lavoster alfo faturated a given quantity of foda with nitrous - acid, but as there was an excefs of acid no ftrefs can be laid on his experiment. I found nitrated foda to attract moifture in a moderate degree. NiITRATED BaRYTES. As barytic earth cannot well be diffolved in nitrous acid without the affiftance of heat, I was obliged to attempt the analyfis of this falt by indire& methods, namely, precipitation by cryftallized foda and vitriolated tartarin, Tue foda I employed contained 15 per cent. of fixed air, and 21,5 of mere alkali; and 100 grains of aerated barytes ignited contains about 21,5 of fixed air. Now I found that +1oo grains of crylallized nitrated barytes were precipitated by 105 nearly of thisyfoda, and that the earth Te after [239 5) : after edulcoration, deficcation and ignition, weighed 70,25 grains nearly ; but at the rate above-mentioned thefe 70,25 grains are re- ducible, deduéting the fixed air to 55,10 of pure barytic earth. On the other hand 108 grains of cryftallized foda contain 23,22 ‘grains of mere foda, and thefe we have already feen are capable of taking up 31,41 of real nitrous acid, therefore by this experiment 100 grains of cryftallized nitrated barytes contain 31,41 of acid and 55,10 of earth; the remainder then is water of cryftallization, = 13,49 grains. AGaIN, 400 grains of cryftallized nitrated barytes were diffolved in 2400 of water, and precipitated by the gradual addition of a folution of vitriolated tartarin; the precipitate which was flowly and difficultly formed and collected weighed after ignition about 88 grains; thefe (at the rate of 33,33 per cent.) contained 29,33 of real vitriolic acid, and confequently 58,67 of mere earth. TakING a mean, then, of thefe two experiments, 100 grains of nitrated barytes contain 56,88 grains of mere earth. LastTuy, 308 grains of native aerated barytes diffolved in 240 of nitrous acid, whofe fp. grav. was 1,451, diluted with 5 times its weight of water in a gentle heat afforded 384 grains of cryftal- lized nitrated barytes, befides a {mall refiduum. Now [ 240 ] Now this acid contains 58 per cent. real acid, and confequently 240 grains of it contained 139,2 of real acid; and if 384 grains of the cryftals contain 139,2, then roo grains fhould contain 36,25. But it muft be confidered-that fome was contained in the mother liquor, fome in the cryftals that were not wafhed, but dried on filtering paper, and fome was difperfed by the heat applied. Tuts experiment alfo gives fome, though not an accurate in- formation of the proportion of earth in nitrated barytes, for 308 grains of aerated barytes (at the rate of 21,5 per cent.) contain 66,22 of fixed air, and confequently 241,78 of mere earth. Sup- pofing then 384 grains of nitrated barytes to contain this quantity of earth, 100 grains of this falt fhould contain 62. But this fup- *pofition is inadmiffible by reafon of the loffes juft mentioned. Upon the whole we may ftate the centenary proportion of this falt at 57 of earth, 32 real acid, and 11 of water. Hence Ioo grains barytic earth take up 56 of real nitrous acid, and fhould afford 175,43 of nitrated barytes. And roo grains real nitrous acid fhould take up 178,12 grains of barytic earth, and fhould afford 312,5 of nitrated barytes. 100 grains of this falt loft only 4 a grain of its weight by ex- pofure to a heat of 300° for half an hour. It is alfo difficultly foluble. Its folution when faturated does not redden Litmus. NITRATED SSE [ 241 ] NITRATED STRONTHIAN. 100 grains of perfectly cryftallized nitrated ftronthian, diffolved in 480 of water, were precipitated by about 107 of cryftallized foda, containing 16 per cent. of fixed air and 21,5 of mere alkali, the precipitate, after ignition, weighed 53,25 grains, and contained 17,04 of fixed air, and confequently 36,21 of mere earth—Alfo the 107 grains of foda (at the rate of 21,5 per cent.) contained 22,9 of mere alkali, which (at the rate of 135,71 per cent.) took up 31,07 of real nitrous acid ; then by this experiment 100 grains cry- ftallized nitrated ftronthian contain 36,21 of earth, 31,07 of acid, and 32,72 of water. THEN Io grains of pure ftronthian earth take up 86 nearly of real nitrous acid, and fhould afford 276 of cryftallized, or about 92 of thoroughly deficcated nitrated ftronthian. AnD 100 grains real nitrous acid fhould take up 116,5 of mere ftronthian earth, and afford 321 of cryftallized, or 107 of thorough- . ly deficcated nitrated ftron:hian. Section 3. Nirratep Lime. In my experiment 136 grains Carrara marble were faturated by 400 of nitrous acid, whofe fp. grav. was 1,2754, and which confe- Vou. Vil. fae Hh es quenily. [ 242 J quently contained (at the rate of 33,59 per cent.) 134,36 real acid.. . h * The 136 grains Carrara marble contained (at the rate of 55 per cent.) 74,8 of lime. CoNsSEQUENTLY Ioo parts lime take up’ 179,5 of real nitrous acid, and 100 parts real nitrous acid take up 55,7 of lime. Lavosier diffolved 972 grains of flacked lime, dried in a heat of about 600°, in 3456 grains of nitrous acid, whofe fp. grav. was 1,2989, and confequently contained (at the rate of 36,7 per cent.) 1268 grains real acid; from the 972 grains lime we muft dedu@ (at the rate of 28,7 per cent. water abforbed in the flacking) 268,9 of water, and alfo 35 grains of fixed air, abforbed while flacking and drying, there remain then 668 of mere lime, and thefe took up 1268 of real acid, then 100 grains of lime would take up 190 of teal acid. 1 Lavofier, 198. Perhaps the difference arifes from my computing the quantity of real acid from a fpecific gra- vity taken at 60°, whereas his might have been taken at a higher degree. Bereman found 1oo grains of nitrated lime, well dried (that is dried in air) to contain 32 of lime; by the above analogy the pro- portion of the other two ingredients may be found, for fince 100 parts lime take 179,5 of real acid, 32 fhould take 57,44, confe- quently the remainder, viz. 10,56 are water; if the nitrated lime could be perfectly dried, it would contain about 36 per cent. of lime and 64 of real acid. AccorRDING [ga3.04 AccorpinGsto Wenzel 122,66 grains of lime take up 240 of his ftrongeft acid, confequently 100 of lime would take up 195,64 of fuch acid, but this quantity is equivalent to only 159 of my real acid, this difference I cannot account for. Sefion 4th. NiTRATED MacGnesta. By my experiment 100 parts mere magnefia require 210 of real nitrous acid for their faturation. AND 100 grains real nitrous acid take up 47,64 of mere magnefia. a 100 grains cryftallized nitrated magnefia contain 46 real acid, 22~- magnefia, and 32 of water, as I found. AccorpincG to Wenzel 77 grains of the magnefia he employed contained but 32,13 mere earth, and yet required 240 of his fp. of nitre for their faturation, which fp. of nitre, by my calculation, contained but 67,2 real acid, and confequently 100 grains mere magnefia would require 209 real nitrous acid ; by his own calcula- tion 240 of his fp. of nitre contained 82,5 of his ftrongeft nitrous acid, and confequently 100 grains mere magnefia fhould take up 2.56 of fuch acid, = 207,87 of my real acid. © Hha2 Accor DING ec eee Accorpine to Fourcroy, 4 An. Chy. 214, 150 grains aerated magnefia; ‘containing 48,66 per cent. mere magnefia, and confe- quently in all 73 grains, were faturated by 222 grains of nitrous acid, whofe fp. grav. appears to have been 1,5298, of which 100, by my table, contain 69,88 real acid, and confequently 222 con- tain 155; and if 73 grains mere magnefia take up 155 real acid, 100 grains mere magnefia fhould take up 212. CHAP. ‘GCHAP. IV. OF MURIATIC NEUTRAL SALTS. Section if. Or Muriatrep TarRTaRIN. In my laft paper I have ftated that 86 grains of mere tar- tarin were faturated by 254 grains of muriatic acid, whofe fp. grav. at 60°) was 1,1466; this is extratabular, but intermedi- ate between the tabulated fpecific gravities 1,147 and’ 1,1414, but nearer to the higher, and its. centenary acid contents will be found by the 1ft problem to be 19,06; confequently 254 grains of this acid liquor contained 48,412 real acid; the fum of, the acid and alkaline parts then amounts to 48,412 + 86, = 134,412 of muriated ‘tartarin; and fince 134,412 of this falt contained 86 of alkali, 100 parts of the dry falt fhould con- tain 64 nearly of tartarin, and the remainder or 36 parts are real marine acid. HENcE 100 grains tartarin take up 56,3 of real marine acid, and fhould afford 156,3 of well dried muriated tartarin. And Io00 grains real marine acid fhould take up 177,6 of mere tartarin, , and afford 277,6 of deficcated muriated tartarin. WENZEL xy [ 246 J Wenzsv found 83,5 grains of tartarin to afford him 129 of muriated tartarin, confequently 100 parts of this falt fhould contain 64,7 of alkali, and 25,3 of acid, and foo parts tartarin fhould take up 54,491 of real acid, and afford 154,491 of mu- riated tartarin, all which determinations differ very little from mine, and afford no inconfiderable proof of the accuracy of the table. Hence we may deduce the quantity of real acid in Wenzel’s fp. of falt and its fp. gravity. By his own account 202 grains of his fp. of fale contained 45,5 of his ftrongeft acid, confequently 100 grains of it fhould contain 22,52, and 240 grains of it 54, and its {p. gravity about 1,174. By my calculation 202 grains of his fp. of falt contained 46,44 . of my real acid, and 100 grains of it contained nearly 23 of my real acid, and 240 of it contained 55,17, and its fp. gravity fhould be about 1,176. . AxoTHER proof of the accuracy of my determinations will be found in the 2d §. Kvarrorn’s determination agrees fully with mine, for to 116 grs. of fylvian he afcribes 42 of concentrated muriatic acid, confe- quently 100 grains of fylvian fhould contain 36,2. 1 Klapr. 134. v = Section eer ee Ll apes] Seéhion 2a. COMMON SALT. Ir has been feen in my laft paper that 30,05 grains of meré foda were faturated by 129 grains of muriatic acid or fp. of falt, whofe fp. gravity in the temperature of 60° was 1,1355; ‘this by the table contains about 17,5 real acid per cent. confequently 129 grains of it contained 22,07 grains real acid, if therefore the neutral falt here formed contained nothing elfe but mere foda and real acid, its weight fhould be 30,05 + 22,07= 52,12. Yet by the laft experiment it appeared that the weight of the falt thus formed amounted to 56,74 gr. the furplus 4,62 grains muft therefore have been water, and fince 56,74 grains of common falt contain alkali, real acid and water in the above proportions, 100 grains of common falt (well dried and deprived of the water interfperfed between its pores) muft contain 52,96 foda, 38,88 real acid, and 8,16 water of compofition that always accompanies the acid when this falt is formed, and therefore muft in all other of examining the compofition of this falt, have been con- founded with it. In this fenfe therefore I may fay that 100 parts common falt contain in round numbers 53 parts alkali and 47 of acid. Hence 100 parts mere foda take up 73,41 of real marine acid, ~ or 88,74 of the aqueous acid, and then afford 188,74 of common falt. And too parts of the aqueous acid fhould take up 1 12,688" of [ 248 J of foda, and afford 212,688 of common falt, and too grains real marine acid fhould take up 136,31 of foda, and afford 257,2 of common falt. 100 grains of the aqueous acid contain 15,33 of water. Accorpinc to Wenzel 131,5 of ignited common falt contain 71,5 of alkali, and 60 of his ftrongeft marine acid ; confequently 100 grains common {alt fhould contain 54,3 of alkali, and 45,7 of that acid. And 100 parts mere foda fhould afford 184 nearly of ignited common falt. This ftatement differs very little from mine, and from Weigleb’s ftill lefs, for he found 100 parts common falt to contain “53,5 of alkali, and 46,5 of acid, and 100 parts foda fhould take. up 87,5 of acid, and afford 187,5 of common. falt. Bur Mr. Bergman’s ftatement differs widely from the fore- going, both Wenzel and I have found the alkaline part to exceed the acid, he on the contrary found the acid to exceed by much the alkaline, for to 100 parts common falt he'affigns 52 of acid, 6 of water and only 42 of alkali. From the’ great refpect I have ever entertained for this excellent man, this ‘circumftance always ” gave me much uneafinefs. ‘To inveftigate the truth by direct ex- periment otherwife than was always done appeared dificult. I therefore endeavoured to difcover it by an zmdireé experiment, namely, by finding how much cauftic foda might be obtained from the decompofition of a-given quantity of common falt. this decompolition ee a ne ee [} 249. J decompofition I effected by tartarin, but the exa@ feparation of the foda from the fylvian was fo difficult that I defpaired of ob- taining fatisfaction in that wy: luckily, however, a more patient and fkilful experimenter, Mr. Hahneman has fince performed this experiment, and found that rr parts mere tartarin were requi’te to feparate 7 of mere foda from common falt.* We may -there- fore now examine with which of the two oppofite {tatements this proportion is beft fuited. By my determination 7 grains foda enter into the compofition of 13,21 of common falt, and this quantity of common falt contains alfo 5,13 grains real acid, which muft be taken up by the tartarin to fet the 7 grains of foda free. Now fince 100 parts tartarin take up 56,3 of real marine acid, 9,12 of tartarin fhould take up 5,12 of this acid, which falls fhort of Hahneman’s refult by 1.88 grains. But it is well known that fomewhat more of any divellent agent mutt be applied to effet an intire /(paration of any principle than would be neceffary to faturate that ee if it were in afree Bneseed f {tate. By Bergman’s determination 7 grains of foda enter into‘the compofition of 16,66 of common falt, and this quantity of common falt contains alfo 8 of real marine acid, now, as’ ac- cording to him 1oo parts tartarin take,up 51,5 of the ftrongeil or real marine acid, 15,53 would be requifite to take up 8 of that acid, which exceeds Hahneman’s refult by 4,53 grains, whereas by the above reafon it fhould rather fall fhort of it. Vo. VII. li Bur * 2 Chy. An. 1797, p- 396 [ 250 ] Bet there are two other experiments which fet the inaccuracy of his determination in a ftill clearer light, the one executed by Mr. Wolfe, aud the other by Dr. Black f. Mr. Wotre found that 120 parts muriated filver or luna cornua, when decompofed by tartarin, afforded 55 grains of fylvian or muriated tartarin ; thefe 55 grains therefore contained all the acid that exifted in 120 of muriated filver. Now Dr. Black found that 235 grains of muriated filver contain all the acid that - exifts in 100 grains of common falt, and confequently 120 grains of the muriated filver contain all the acid that exifls in 51,06 of common falt, whence it follows that 55 grains of fylvian and 51,06 of common falt contain the fame quantity of acid, fince the firft received and the latter gave out all the acid that exifts in 120 parts muriated filver. WE may now fee in which of the 2 different ftatements this . equality is found, or whether in neither or in both. 1ft. Accorpinc to Bergman Ico grains of muriated tartarin * contain 31 of real acid, then 55 grains of that falt fhould contain 17,05. AGAIN, Ico orains of common falt contain by his ftatement 52 of real acid, then 51,06 of this fal: fhould contain 27,55; thefe quantities are evidently very diftant from an equality. + Phil. Tranf. 1776, p. 611. 3 Edinb. Tranf. 116. a i a Pear : _ad. By my ftatements 100 parts fylvian contain 36 of real acid, then 55 parts of this falt fhould contain 19,8; alfo 100 parts common falt contain 38,88 real acid, then 51,06 parts of this falt fhould contain 19,85. | . Seéfion 3d. Mouriatep BAaryYTEs. Tue proportion of ingredients in this falt may be inveftigated from the following fadts : ift. Kraprotu found that 73 grains of aerated native barytes (which contained an inconfiderable proportion of ftronthian) fa- turate 100 grains of muriatic acid, whofe fp. grav. was 1,140 di- luted with 200 grains of water, and that Ioo grains of aerated barytes contain 22 of fixed air, 2 Chy. An. 1793, p. 195 and 146, and 1: Klapr. 269, therefore 73 grains of aerated barytes contain 56,94 of barytic lime: . adly, He found that 56,59 pure aerated barytes diffolved in this acid afforded 6850 of cryftallized muriated barytes. 2 Klapr. 84. THEN I00 grains of aerated barytes, or 78 of mere barytes, would give 21,04 of muriated barytes. And 100 grains of mere barytic earth fhould give 155 nearly of cryftallized muriated barytes. : AccorpinG to Fourcroy, 4 An. Chy. 71, foo grains native _barytes afford but 112 of deficcated muriated barytes; yet Pelletier ie: tells f esac] tells us, that 100 grains native aerated barytes afforded him 138 of cryftallized muriated barytes, but moft probably it retained fome of the mother liquor. Hence I -deduce, 1ft, that as 100 grains muriatic acid, 1,140 contain 18,11 real acid, 56,94 of barytes tock up that quantity. ConsEQuEeNntLy 100 grains mere barytic earth take up 31,8 of real marine acid, and afford 155 of cryftallized muriated barytes. Anp 100 grains real marine acid fhould take up 314,46 of barytes. WE may alfo remark, that the muriatic acid whofe denfity is 1,140, being mixed with twice its weight of water, will have its fp. grav. 1,0427 which is nearly the fame as that which Fourcroy ~ found beft adapted to fuch folution, namely, 1,0347; and perhaps if the temperature were equal would approach each other fill more nearly. It appears then that the real acid fhould be accom- panied with 16 times its weight of water. adly, Ir follows, that 121,04 parts cryftallized muriated barytes contain 78 earth, 24,8 acid, and 18,24 water, confequently roo parts of the cryftallized falt contain 64,44 earth, 20.45 acid, and 15,06 water. Ann 100 grarns of the deficcated contain about 70 of earth, 22 of acid, and 8 of water. (PER Ee L453] (Per Crawford, quoted by Schmeiffer in Phil. Tranf. 179, 421, muriated barytes is nearly as foluble in hot as in cold water, and three times lefs foluble than muriated {tronthian.) To confirm this conclufion I muft add, that having precipitated a folution of roo grains of cryftallized muriated barytes by a folution of nitrated filver, I found the precipitate duly dried to weigh 118 grains, which as we fhall prefently fee argues the pre- fence of 19,51 of real marine acid. I alfo found that roo parts muriated barytes expofed to a heat of 300° for two hours, loft 16 grains of water of cryftallization, hence we may rate in round numbers the proportion of ingredients in this, falt, at 64 of earth, 20 of acid, and 16 water of cryftallization. Section 4th. MuriatTep STRONTHIAN. Kuaprotu obferved, that 55 grains of native mild ftronthian faturated 100 of marine acid, whofe fp. grav. was 1,140, this being diluted with so grains of water, 100 grains marine acid of this fp. grav. contain, computing from my table 18,11 grains of real acid, and 55 grains mild ftronthian, (at the rate of 69 per cent.) contain 37,95 of mere earth. HeNncE TE oper] Hence I conclude, that too grains mere ftronthian earth take up 47,79 of real acid (fince 37.95 take 18,11 of real acid) and would afford, as we fhall prefently fee, 254,84 of cryftallized mu- riated ftronthian, or 147,79 of deficcated ftronthian*. Anp 100 grains real marine acid enter into the compcfition of 209 grains of deficcated ftronthian, or of 360 of the cryftallized. Acain, Dr. Hope found, that too grams cryftallized murtated Jfironthian contain 42 of water of cryftallrzation, and confequently 58 of deficcated which contain earth and acid in the proportion above mentioned (or 100 earth to 47,79 acid) that is, 39,24 of earth and 18.76 of acid, this proportion agrees very exaétly with that obferved by Pellitier+, for he found 100 grains of native ae- rated ftronthian (which contain 69 of earth) to afford 176 of ¢cry- ftallized muriated ftronthian. Anp fince, in Dr. Hope’s experiment, 39,24 of this earth af- forded roo grains of muriated ftronthian, 69 fhould afford 175,8. Some experiments however of Mr. Lowitz vary confiderably from the above flatements, it app: ared to him that in muriated ftron- thian the quantity of acid exceeded that of earth in the proportion of 54 to 46f; if fo, 100 grains of muriatic acid of the fp. grav. 1,140 fhould contain 44,54 of real acid, for it took up 37,95 of earth * 9 Chy. An. 1793, p- 194. +21An.Chy. p.128. + rft.Chy. An. 1796, p. 128, 129. oe earth in Klaproth’s experiment already quoted, which is incon- fiftent with the proportion of real acid. | have found in muriatic acid ina multitude of experiments, and contrary to all analogy, as we fee that by barytes and fixed alkalis betwixt which this earth undoubtedly ftands, take up lefs than their own weight of real marine acid; it is alfo contradi¢ted by Pellitier’s experiment, for fince 100 grains native aerated ftronthian contain 69 of earth, thefe at the rate of 46 to 54 fhould take up 80 grains of real mu- riatic acid, and the fum of both would be 149 grains: and fince: ‘by Dr. Hope’s experiment 58 grains of united earth and acid take 42 grains of water of cryftallization, 149 grains fhould take 107; and hence inftead of 176 grains of cryftallized muriated fironthian we fhould have 256 grains from 100 of aerated ftron- thian. KiLAprota informs us, that from a folution of 100 grains of aerated ftronthian in muriatic, precipitated by the addition of con- 'centrated vitriolic acid, as long as any precipitate appeared, he ob- tained no more than 114 grains of vitriolated fironthian, and that dried only in air*; whereas the precipitate fhould amount, if the whole of it were obtained, to 118 grains; for fince 58 grains of this earth, as he elfewhere relates, + afford 100 of vitriolated fironthian, 69 fhould afford 118; it is plain therefore that the ma- rine * 2 Chy..An. 1793, p» 200. : + 2 Klapr. p. 97. [ 256 ] rine acid retained fome, or that a fufficiency of the vitriolic acid was not added. This earth isnot therefore a proper teft of vi- triolic acid, at leaft not as proper as the barytic. To obtain a lefs circuitous proof of the proportion of ingredi- ents in 100 parts of this falt, I precipitated a folution of 100 grs. of cryftallized muriated ftronthian by mild foda; the precipitate after ignition weighed 56,75 grains, but thefe being diffolved in marine acid gave out 17 grains of fixed air, and therefore contained only 39,75 of mere earth. ‘ adly. I precipitated a folution of another 100 grains of this cryftallized falt by a folution of nitrated filver, and found the pre- cipitate duly dried to weigh 110 grains, a weight which indicates the prefence of 18,19 grains real marine acid. The weight of the 3d ingredient, namely water, muft therefore amount to 42,06 grains nearly, as Dr. Hope has ftated. Hence we may rate the proportion of ingredients in 100 parts of this falt at 40 of earth, 18 of acid, and 42 of water. And to 1oo parts of the deficcated fait we may allow about 69 of earth and 31 of acid. Hence 100 parts fironthian earth take up 45 or more, exactly 46 of real marine acid, and fhould afford 250 of cryftallized, or 145 of deficcated muriated ftronthian. And 100 farts real marine acid ae: ae acid fhould take 222, or more exactly 216,21 of fironthian earth, and afford 540 of cryftallized, or 313,5 deficcated muriated ftron- thian. Seftion sth. Mouriatep Lime. In my experiment already mentioned 158 grains of powdered Carrara marble were faturated by 402 of muriatic acid, whofe {p. grav. was 1,1355, which contained 17,5 per cent. real acid; therefore 402 grains of it contained 70,55 real acid. The 158 grains marble (at the rate of 53 per cent.) contained 83,74 of lime. Then 83,74 grains lime took up 70,55 of real marine acid. To effect a faturation a heat of 1609 was employed to- wards the end of the folution. ‘Hence 100 grains of lime would faturate 84,488 of real marine acid. And 100 grains real marine acid would faturate 118,3 of ‘lime. In Wenzel’s experiment the acid was not faturated, and hence the refult differs from that of mine. To 240 grains of his fp. of falt he added 120 grains of fragments of purified oyfter-fhells (which, as we have already feen in treating of felenite, con- tained 52,8 percent. of lime,) and at the latter end expofed them Vou. VII. Kk to [ 258 ] to a gentle heat, and when no fenfibie folution appeared he fepa- rated what remained undiffolved, and found that after wafhing and drying it, it weighed 19,625 grains; hence he concluded that 100,375 grains of thefe fhells were diffolved; but then he had no reafon to think the acid was faturated, or that in a longer time it would not take up more, efpecially as the fhells were not in a fine powder, nor did he apply any teftasI did. Having eva- porated the folution to drynefs and heated the dry mafs to fufion, he found it to weigh whilft ftill red hot 106,125 grains. Tuis fhews the folution not to have been faturated, for 100,375 grains of the fhells contained 53 of lime, and the 240 grains fp. of falt contain 54 of real acid by his own account , therefore, as faturated muriated lime lofes no acid in a melting heat, the falt fhould weigh even by his eftimation 107 grains, and by my calculation 112 grains; the remainder therefore of the unfaturated acid was expelled by the heat of fufion. Accorptneé to him 100 grains lime fhould take up 1o2 grains of the ftrongeft marine acid. Ir muft be remarked, alfo, that this falt though in a melting heat ftill retains fome water, and Wenzel’s experiment fhews how much; for by my determination 53 grains lime take up only 44,75 of real acid; and the fum of the ingredients in Wenzel’s experiment amounts only to 97,75 grains ; yet he found the weight 106,125; then 8,375 grains were water. THEN —— ee a eee a 7 eae % [* 250° -] THEN 100 grains muriated lime, weighed red hot, contain nearly 50 of lime, 42 of acid and § water. Berean agrees with me fo far as fltating the proportion of lime in this falt to be fuperior to that of acid; to too parts of this falt he afligns 44 of lime and 31 of acid, but the proportion of acid is higher, for to 44 of lime 37 of acid appertains, by’ the proportion above ftated then that of water is 1g. Note. His falt was weighed at far a lower temperature than Wenzel’s, and hence the quantities but not the proportions in 100 9 q prop grains of it are altered, as it powerfully attraéts water. Section 6th. MoriatTep Macnesia. THE proportions of acid and bafis in this falt are difficultly deter- mined, as it powerfully attracts moifture and eafily lofes its acid iff ftrongly heated, and without fuch heat will retain much water. In my experiments it appeared that 100 grains mere magnefia took up 215,8 of ftandard, or 111,35 of real marine acid. Ayp 100 grains real marine acid take up 89,8 of mere mag- — Nefia. Kka2 KLAPROTH [ » 260%] Kiarrotu * found 420 grains of muriated magnefia evaporated to drynefs to contain 290 of magnefia; as it was precipitated by foda he probably meant mild magnefia, which generally contains but 0,45 of earth; if fo, 290 contained but 130,5 of mere mag- nefia ; confequently 100 grains of muriated magnefia gently but fenfibly dried fhould contain 31,07 mere magnefia, and this by my computation fhould take up 34,59 of real acid. The remainder is therefore water. WeNzeEL’s experiments accord with mine with refpe@ to the fu- periority of the proportion of earth to that of acid in a given weight of muriated magnefia. According to him 100 grains of mere magnefia take up 122 of real marine acid; but by my com- putation of the quantity of real acid in his fp. of falt, v. zs. 23 per cent. allowing his mild magnefia 45 per cent. of earth, 100 grains of it fhould take up 115,8 real muriatic acid. BERGMAN’s refults differ from thefe very widely, for according tohim 41 grains mere magnefia take up only 34 of the ftrongeft ’ marine acid. Section - * x Klapr. 369. . 4 Pan Section 7th. MourtaTep SILVER. Ir isnow well known from the experiments of Margraff, Berg- man, Klaproth, Wolfe, Wenzel, &c. to which I need not add my own, that 100 grains of muriated filver contain very nearly 75 of filver when dried in a heat of 80°, or 75,235 when heated more but not fufed, as in Wenzel’s and Wolfe’s experiments; butit muft not be inferred that the remaining 25 grains ae mere. marine acid, for filver diffolved in nitrous acid takes up 10,8 per cent. of oxygen ; therefore 75 grains of it take up 8,1, which fubftraGted from 25, leaves the quantity of acid 16,9; or if the muriated filver were much heated, the acid and oxygen would amount only to 24,76; and deduéting the oxygen, the acid fingly would be 16,6 grains; this agrees exa@ily with Wolfe’s experiment, for he found as al-- ready faid that 120 grains of this metallic falt decompofed by tar- tarin afford 55 of muriated tartarin. Now 120 grains contain by this computation 19,92 of real acid; and as 100 grains muriated - _tartarin contain 36 of real acid, 55 grains of it fhould contain 19,8; the difference is infignificant. Hence 100 grains filver take up 22,133 of real marine acid, and afford 133 of muriated filver by the addition of oxygen. Anp 100 grains real marine acid unite to.451,87 of filver, and: afford 602,4.0f muriated filver. - | 100 [ hear 100 grains pure cryftallized common falt precipitate from a folu- tion of nitrated filver 233,5 grains of muriated filver by Klaproth’s, 235 by Dr. Black’s, and 237 by Arrhenius’s experiments*; Dr. Black’s is a medium between both; the difference arifes only from the degree of deficcation. 100 gtains of muriated tartarin fhould produce 216,86 of mu- riated filver. Seétion 8th. MuriatTep Leap. Tuts falt may be obtained in two ftates, either in acicular cryf- tals or thoroughly deficcated. ‘The proportion of ingredients in each I deduce from the following facts : 1ft. Kuaprota having diffolved roo grains lead in dilute nitrous acid, and precipitated the lead by cauftic tartarin, found the precipitate fharply dried until it began to grow yellow, to weigh 115 grains. 1 Klaproth, 274. 2d. Havine precipitated a folution of 100 grains of lead in nitrous acid by dropping muriatic acid as long as any precipitate appeared, and evaporated the whole to drynefs in a fand heat, he found the muriated lead to weigh 133 grains. Ibid. 3d. * Mem. Stock. 1785. [e634 3d. He alfo found that 22,5 grains of cryftallized acicular - muriated lead, well drained and dried by expofure to the air, contained 16 grains of metallic lead, therefore 100 grains of fuch cryftals fhould afford 71,11 of metallic lead. First, to thefe facts I muft farther add, that in muriated lead, whether cryftallized or deficcated, the lead is in a calcined ftate. Hence I infer, that fince 100 grains of metallic lead give 133 of calx of lead, the 71,11 grains of metallic lead in 100 parts cryftallized muriated lead amount to 81,77 of calx of lead. The calx, including not only the metallic lead, but alfo oxygen and water, as we fhall prefently fee; the remainder therefore is real marine acid, amounting to 18,23 grains. AGAIN, as 133 grains of the thoroughly deficcated muriated lead contain 100 of metallic lead, 100 grains of this muriated lead muft contain 75,12, but 75,12 metallic lead form 83 of . calx; the remainder therefore muft be real marine acid = 17 grains. Tuese conclufions are farther confirmed by the experiment of Mr. Wolfe. Phil. Tranf, and 1o Roz. 370. Having decom- pofed 120 grains of muriated lead dried by expofure to the air by a fufficient quantity of tartarin, he found them to produce 61 grains of muriated tartarin. Therefore both the 120 grains mu- riated [ 264 | riated lead and the 61 grains of muriated tartarin fhould contain the fame quantity of real marine acid. Now if too grains mu- riated lead dried in air contain 18,23 real acid, 120 grains of it fhould contain 21,87 real acid. Anp fince 100 grains muriated tartarin contain by my former determination 36 grains real acid, 61 grains of this falt fhould contain 21,96; the difference is only 0,09 of a grain. As tothe 115 grains calx of lead produced in the precipitation of a folution of roo grains of lead in nitrous acid by cauftic tartarin, I have already fhewn in the 2d vol. of my Mineralogy, p. 497, that 100 parts lead, when diffolved in nitrous acid, take up 5,8 of oxygen*, therefore the remainder is water, = 9,2 grains. HENCcE 100 parts metallic lead take up about 25,63 of real ma- rine acid, and afford 140,62 of cryftallized muriated lead, or 133,12 of the deficcated. Anp 100 grains real marine acid unite to 394,06 of metallic lead, and afford 548,64 of cryftallized muriated lead. AND * Fourcroy, 2 An. Chy. 213, ftates the quantity of oxygen at 12,5 in 100 of muriated lead, but this is contradicted by the experiment of Mr. Wolfe, &c. He moft probably means the muriated lead formed in the folution of a calciform ore. Fie a Se 7 - [ 266. ] Anp too parts cryftallized muriated lead contain 81,77 calx of lead (= 71,11 metallic lead,) and 18,23 of real marine acid, Anp too grains thoroughly deficcated muriated lead contain 83 calx of lead (= 75,12 metallic lead,) and 17 of real marine acid. AccorpincG to Wenzel, roo grains metallic lead fhould afford 137,5 of deficcated muriated lead; he probably dried it fome- what lefs than Klaproth had done. The proportions of lead and acid he could not well determine, the exiftence and proportion of oxygen not being known when he wrote. Note. The quantity of metallic lead obtained from too parts cryftallized muriated lead by fufion with black flux is much fmaller than that above ftated. (fee 1 Klapr. 171,) as much is retained by that flux. Yet fee 3 Weftrumb. Phyfical and Chem. Abhandl. 383. Or AERATED VoL-ALKALI aND AMMONIACAL SALTS. Tue former experiments which I made with a view of afcertain- ing the proportion of ingredients in thefe falts were defective in feveral refpeds : Vou. VIL. Ll ft [i ages" =} ift. For want of a due eftimate of the quantity of mere vol- alkali in a given quantity of aerated alkali, the fubftance to be fa- turated with the three other mineral acids. Dr. Prieftly’s experi- ments, the bafis of the eftimate I then formed, not exhibiting the temperature and preffure of the atmofphere when the volumes of fixed and alkaline airs were combined, afforded an opportunity for forming rather an approximation than an accurate determination of their feveral weights. ad. J was not then aware of the difficulty of finding the exaé point of faturation of the aerated vol-alkali with the mineral acids ; a difficulty however mentioned by Macquer*, and fo great that Du Hamel judged it impoffible to vanquifh itt. Wenzel very fa- gacioufly abforbed the excefs of acid by oyfter fhells, but in my mode of experimenting this teft could not be applied; hence there was an excefs of acid in all of them. ‘Thefe errors induced me to analyze rather than compofe thefe falts. Or ArraTep Vo.L-ALKALI. By diftilling roo grains of aerated vol-alkali with 300 of dry flacked lime in a pneumatic apparatus and a fand heat I obtained 129 cubic inches of alkaline air, barometer 30,2, and thermometer at * Macquer’s Elem. 389, Englith. + Mem. Par. 1735, p. 664, in 8vo. [267° J at a medium 62,5. 100 grains of alkaline air weigh 18,16 grains, as I have fhewn in a former treatife, barometer 30, thermometer 61. Then at that barometrical height 129 cubic inches would be- come 130; but as the heat in the prefent experiment exceeded 61, the expanfion refulting from it muft be fubftra&ted ; and according to Mr. Morveau, 2 An. Chy. a volume of this air at 32° being taken as 1 becomes at 77° 1,2791, and confequently gains 0,0062 by each intermediate degree, confequently the volume of this would at 61 be only 129,1; its weight therefore is nearly 24 grains. ‘This falt contained 52 per cent. of fixed air, confequently its ingredients were 52 grains fixed air, 24 of mere alkali, and 24. of water. ; The proportion of vol-alkali in aerated vol-alkalis vary, increafing or decreafing with the proportion of fixed air they contain. Mr. Cavenntisu in the Philofophical Tranf. for 1766, p. 169. found that 1643 grains of aerated vol-alkali, containing 53,8 per- cent. of fixed air, faturated the fame quantity of marine acid as 1680 of another parcel, which contained but 52,8 per cent. of fixed air. Hence the quantities of mere alkali in each were reciprocally as 1680 to 1643, and thefe are nearly to each other as 53,8 to 52,8; __ and as the aerated vol-alkali that contained 52,8 per cent. of fixed air contained 24 per cent. of mere vol-alkali; that which contained 53,8 per cent. of fixed air fhould have contained 24,83 per cent. Lla HENcE ogee et] Hence the proportion of fixed air in aerated vol-alkalis is to that of mere alkali in thofe falts as 13 to 6, and the remainder is water of compofition. WENZEL, p. 100, alfo perceived that the proportion of mere alkali in aerated vol-alkali was very fmall, and ftates it nearly as low as Ido; for to 240 grains of this falt, containing 53,75 per ; cent. of fixed air he afcribes 129 of fixed air, 31,125 of water, and confequently 79,875 of mere alkali. Hence 100 grains fhould contain 53,75 fixed air, 33,28 of alkali and 12,97 of water. ComMMon Sat AMMOoNIAC. By diftilling in a pneumatic apparatus and a fand heat, roo grains of fublimed fal ammoniac and 300 grains of quick lime, I found it to yield as much alkaline air as amounted to 25 grains, with fome few drops of water; the remainder of the water being pro- bably detained by the lime or by the muriated lime which is known to retain water moft obftinately. By treating 100 parts of this falt in folution with a folution of nitrated filver, I found it to afford 258,5 of muriated filver heated to fufion, and confequently to contain 42,75 of real marine acid. Hence [ 265 ] Hence too parts of this falt contain 42,75 of real marine acid, 25, or making allowance for loffes, 28 of mere vol-alkali, and 29,25 of water of cryftallization and compofition. _ Hence too parts mere vol-alkali take up 152,68 of real marine acid, and fhould afford, if there were no lofs, 357,14 parts of fublimed fal ammoniac. And 100 parts marine acid take up 65,4 nearly of mere volalkali, and fhould afford 233,9 parts of fublimed fal ammoniac ; but in fubliming fal ammoniac there is always fome lofs. _ Mr. Cavendifh, in the Philofophical Tranfations for 1766 tells us, that 168 parts aerated vol-alkali, containing 52,8 per cent. of fixed air, faturated as much marine acid as 100 grains of marble, which contained 40,7 per cent. of fixed air; now 100 grains of this marble contain, by the analogy formerly given, (45 of fixed air to 55 of lime) 50 grains of lime, by the 2d table, take up 42,2 of real marine acid, and 100 grains of the aerated vol-alkali there men- tioned, contain 24 per cent. of mere alkali, and confequently 168 grains of it fhould contain 40 of mere alkali, which by the above ftatement would require for faturation 61 of real marine acid. This experiment would have made me doubt of the propriety of the above conclufions, had not Mr. Cavendifh exprefsly ftated that his foiution of marble was faturate, (and confequently as a fatu- rate folution cannot be obtained without heat, which he did not apply, he muft have added an excefs of marble, and judged the folution [ 270 ] folution faturate when no more air was expelled) and on the other hand he tells us, that the alkaline folution contained an excefs of acid, and this excefs exifling in every particle of a large folution muft be confiderable. In the experiment related in my laft paper, I ftated that roo grains of aerated vol-alkali were faturated by 246 of marine acid, whofe fp. grav. was 1,1355, which appears by the firft table to contain 17,5 per cent. real acid, and confequently the quantity in 246 grains was 43 grains; on the other hand, the vol-alkali, con- taining but 43 per cent. of fixed air, contained, by my actual ex- periments, only 19385 grains of mere alkali, and this quantity fhould take up but 30 of real marine acid. Hence in my former experiments there was an excefs of 13 grains of acid, which made the fp. grav. equal to that of the teft folution, and thus induced me to think the quantity of fal ammoniac formed greater than it really was. WenzEL found 168,4 grains of vol-alkali, containing 53,75 per cent. of fixed air, to require 240 grains of his fp. of falt to faturate them, and this quantity of his marine acid we have already feen to contain §5,17 of real acid, and 168,4 of the aerated alkali contain- ed, by the analogy already ftated, 41,71 of mere vol-alkali, the fam of both was 96,88 ; yet having evaporated the folution to drynefs, and expofing the refiduum to a heat of 2129 for four hours, he found [\ een found it to weigh 110,125 grains, as he knew 56 of thefe to be acid (or according to him 54), he naturally fuppofed the remainder to be vol-alkali ; hence according to him 100 parts of fal ammeniac thus dried contain 49 parts of acid and 51 of vol-alkali, The dif- ference between us feems to arife from the lofs always experienced during evaporation, and if this had not happened, the dry refiduaum would have amounted to 128 grains; as to the quantity of vol-al- kali he had no method of eftimating it. Cornetre perfefly decompofed 2304 grains of fal ammoniac by an equal quantity of lime, which he flacked after weighing it, examining the refiduum, he threw it on a filter, and edulcorated it with repeated effufions of water, and what remained undiffolved he found to weigh, when dry, 756 grains, and hence he judged the remainder, viz. 1548 grains to have been diffolved by the acid of the fal ammoniac, and to confirm this conclufion, he precipi- tated the folution which had paffed the filter with a fixed alkali, and drying the precipitate, found it to weigh 1542 grains* , whence it feems to follow, that the acid contained in 2034 of fal ammoniac had diffolved 1542 of lime, whereas, by my calculation, it fhould diffolve but 1272,46 of lime, for fince too grains of fal ammoniac contain 42,75 of real marine acid, 2304 fhould contain 1008 ; and fince by the third table 100 grains real marine acid take up §18,3 of lime, 1008 fhould take up but 1272,46 of lime. Bur * Mem. Par. 1786, p. 533. IL eRe ea Bur the lime I ufed was pure and perfedily free from fixed air ; can that be faid of the common lime of Marly, which he employed and does not fay he had prepared? Befides, by his edulcorations, much pure lime muft have been diffolved, and have mixed with the folution of muriated lime, and if his alkali were not cauftic, the quantity of lime precipitated by it muft have been at leaft par- tially aerated, and confequently the mere earthy part apparently greater than it would have been if pure. However, as this expe- riment forms a cumulative proof both of the proportion of acid contained-in fal ammoniac, and of the quantity of it taken up by a given weight of lime, I thought it incumbent upon me to repeat it, hence I mixed 50 grains of fal ammoniac with 150 of flacked lime, and heated the mixture in a large glafs phial until all the al- kali was driven off and the mixture ceafed to {mell, I then added a fufficient proportion of water, and digefted the whole in a gentle heat for fome hours, then filtered and edulcorated the mafs on the filter, as I judged the folution to contain lime as well as muriated lime, I paffed a fiream of fixed air into it, which inftantly turned it milky, and then filtered it off; the folution now free from lime I precipitated by a folution of an aerated foda, which contained 17 per cent. of fixed air, as much of the folution was requifite as contained 123 grains of foda. The precipitate collected, edulco- rated and dried for fome hours on the filter, in a heat of 150%, weighed 46,75 grains, though no more could be feparated than 41,62, thefe after ignition weighed 35 grains, fome ftuck to the glafs Lf 993] glafs, and 5,25 remained in the filter; 123 grains of the foda gave out 20,91 of fixed air, and, as I afterwards found, kept about a a grain of the lime in folution, now 21 grains of fixed air are ab- forbed by 23,44 of lime>this then was the quantity of lime taken up by the acid contained in 50 grains of fal ammoniac, that is, 21,37 real marine acid, whereas by my calculation, fince 100 grains marine acid take 118,3 of lime, 21,37 fhould take up 25,28, the difference is 1,84 grains, and even this I believe to proceed from the whole of the fal ammoniac not having been decompofed, 19,8 grains of the acid appear to have been taken up by the lime, and about 3,6 of the ammoniac efcaped decompofition, this alfo clearly appears by te action cf the foda, for 100 grains of this foda con- tain 22 of mere alkali, then 123 grains of it contains 27; as 100 grains mere mineral alkali take up 73,41 of marine acid, then 27 fhould take up 19,82. Hence we fee that in Rigour roo parts fal ammoniac may be decompofed by 1co parts chalk, for 100 parts chalk gene- rally afford 42 of fixed air, and confequently contain 51,3 of lime, and 100 parts fal ammoniac contains 42,75 real acid, and fince 100 grains real marine acid are faturated by 118,3 of lime, 42,75 of this acid require but 49,57 of earth; but in all fuch cafes the medium of decompofitionds always taken in greater quantity than is abfolutely requifite, otherwife the mixture would never be perfe&, and in this cafe part of the falt might fublime without de- compofition ; hence 200 parts chalk are moft commonly ufed, RVs VAT. . Mm . though [yeas though 125 are faid to be fufficient. Doffe Elab. laid open t10, 1 Labor. in Grofs 68, in note per Weigleb, and in effe@ 125 grains chalk, at the above rate, would furnifh 52 grains of fixed air, which would faturate 24 of vol-alkali, and the ammoniac contains a fufficiency of water. Hence alfo we fee how it happens that too parts fal ammo- niac decompofed by 200 parts chalk frequently afford 89, nay, ac- cording to Baumé, even 94 parts aerated vol-alkali, for if there were no lofs 125 parts of chalk were fufficient, but then this large quantity of fixed air is expelled, not by the acid of the fal ammo- niac, but by the heat applied, as Pellitier de la Sale has no- ticed, 2 Pharmacopie de Londres 427, and on this account mag- nefia, as it parts with its fixed air much more eafily, and con- tains more water, affords a quantity of aerated vol-alkali, when ufed as a medium for decompofing fal ammoniac, nearly double that of the fal ammoniac employed. Thus Weftrumb from roo grains of fublimed fal ammoniac and 300 of magnefia obtained 193 grains aerated vol-alkali, 2 Chy. An.1788, p. 15; his magnefia — muft have contained a very large proportion both of fixed air and water, for he fays that 1920 grains of it being calcined left only 600 of earth, ibid. 17. Hence alfo, Dolfuz having treated 100 parts fal ammoniac with 125,,and even with 200 of chalk, in a glafs retort, obtained no more than 50 of aerated vol-alkali; the fame thing happened when he —— se [ as J he ufed an earthen retort, as he fimply heated it to rednefs, whereas a ftrong white heat is requifite to expel fixed air from chalk, 2 Crell. Beytr. 199. I believe unpurified fal ammoniac would yield more aerated vol-alkali than the purified, on account of the oil it contains, which affords fixed air. Another certain proof that 125 erains chalk are not aéted upon by the acid contained in the too parts fal ammoniac, but contribute to the increafed quantity of aerated alkali merely by the fixed air expelled from them by heat, is that the refiduum contains fome calcareous earth which the acid had not attacked, as Richter has obferved, 1 Stock. 2 Theile 98 -and 99. SEVERAL important deductions may be deduced from the know- ledge of the compofition of fal ammoniac, for inftance, an eafy ex~ planation of its great refrigerating power, &c. which being impro- per for this place, I omit. Virriotic AMMoNIAC. 100 grains of cryftallized vitriolated vol-alkali and 300 dry flacked lime, pneumatically diftilled in a pneumatic apparatus and a ftrong fand heat, Bar. 30,2, Therm. 66°, afforded 78,41 cubic inches of alkaline air, = 14,24 grains. Mma From [ 276 jj From a folution of vitriolated vol-alkali, precipitated by a folu- ~ tion of muriated barytes, 164 grains of ignited barofelenite were obtained, hence the falt contained 54,66 grains real vitriolic acid. HENcE 100 grains vitriolated vol-alkali contain 14,24 of mere vol-alkali, 54,66 of real acid, and 31,1 of water. Ix my former paper I ftated the quantity of vitriolic acid in 100 grains of cryftallized vitriolated vol-alkali to be 62,47 ftandard, = 55,7 real acid, the variation is not confiderable, but of the al- kali I could not then form a proper eftimate. HENCE Ioo parts mere vol-alkali take up 383,8 of real vitriolic acid, and afford 702,24 of vitriolated volalkali. adly, 100 parts real vitriolic acid fhould take up 26,05 of mere vol-alkali, and afford 182,94. of vitriolated vol-alkali. Accorpine to Wenzel, alfo, 100 parts vitriolic ammoniac con= tain 58,8 of real acid, hence of all cryftallized falts it contains the greateft proportion of this acid, as Glauber does the leaft. NitrratTrep VoL-ALKALI. From 50 grains of cryftallized nitrated vol-alkali, mixed with twice its weight of flacked lime, I obtained, in a pneumatic ap- paratus, 40 cubic inches of alkaline air, Bar. 30,06, Therm. 61°, by [ Page: VP by the fimple heat of a candle; fome water alfo paffed, which un-- doubtedly abforbed fome air, a greater heat could not be applied without rifking a decompofition of the alkali itfelf; hence 100» grains of this falt would yield 80 of air, which in thefe circum- ftahces would weigh 14,52 grains. In another experiment I ob- tained ftill lefs of this air, for 50 grains of this falt afforded enly 34,962 cubic inches, the barometer indeed flood higher, namely at 30,26, and the thermometer only at 58. Finprnc this method inadequate to the difcovery of the exact quantity of vol-alkali in this falt, Itried the effe@Q of fpontaneous evaporation ona mixture of this falt with lime and water, but: foon found the quantity evaporated fo great that it was very evi- dent it did not proceed from the mere volatilization of the alkaline part, but in a great meafure from that of the water alfo, hence I was obliged to content myfelf with dete€ting the proportion of the acid part. For this purpofe I made a folution of 400 grains cryftallized ni-- trous ammoniac, and to this added a {mall proportion of a folu- tion of tartarin flightly aerated ; as the point of faturation could not be afcertained by any teft, I added but little of the tartarin, , and fet the liquor to evaporate in a very gentle heat. The next day I found fome cryftals of nitre, which I carefully picked out, wafhed and dried, then added more tartarin to the mother liquor, . fet it to evaporate and. cryftallize as before. Thus I proceeded for feveral. [aye feveral days and at laft obtained 412 grains cryflallized, well dried nitre. Now 412 grains nitre contain, by my account, 181,28 grains real nitrous acid, this quantity therefore exifted in 400 grains cf the nitrous ammoniac, confequently 100 grains of this falt fhould.contain 45,3 of real nitrous acid. THERE are however ftreng reafons to think that this falt con- tains much larger proportion of acid; for in the firft place the falt volatilizes without decompofition with the water that holds it in folution, as Berthollet obferved in an experiment I fhall pre- fently relate, and confequently it is reafonable to fuppofe that fome efcaped this way in my experiment, and moreover nitre is itfelf in fome meafure volatile during the evaporation of its folu- tion, and laftly, both Wenzel, Cornette and myfelf found a larger proportion of acid taken up by vol-alkali during the combination -of both. In my laft paper I ftated the proportion of ingredients in nitrous ammoniac at 24 vol-alkali 78,75 ftandard, which quantity is equivalent to 57,8 grains real acid, but noticed that there was’ an excefs of acid. At prefent all due corrections made from this experiment, I infer that 100 grains cryftallized nitrous ammoniac contain 57 nitrous acid, 23 of vol-alkali and 20 of water. Hence 100 grains vol-alkali take up 247,82 of nitrous acid, and fhould afford 435 of cryftallized nitrated vol-alkali, if there were no lofs in evaporation or no decompofition. AND er ee C505 _.—— | [, 276° J ANp 100 grains nitrous acid fhould take up 40,35 of vol-alkali,, and afford 175,44 of ammoniac, if no lofs &c. Aw experiment of my own, related in my laft paper, feems to contradict thefe refults, for I there ftated that 200 grains aerated. vol-alkali, which contained 50 per cent. of fixed air, and confe- quently the whole, 46 of vol-alkali, having been faturated with nitrous acid, to have afforded 296 of nitrated ammoniac, whereas by calculating from the above ftatements they fhould afford but 200: but the reafon is, that the mafs of falt then procured was not wholly cryftallized, but contained much of the mother liquor and an excefs of acid which increafed its weight. The only ob- jet I had then in view was to fhew that the weight obtained was lefs than could be expected from the theory | had then formed; for this purpofe it was not neceflary to pufh the deficcation very: far—a decompofition alfo took place as will prefently be feen. Accorpine to Wenzel 240 grains of dry uncryftallized ni- trated vol-alkali contain 155,9 of his ftrongeft acid, 77,5 mere vol- alkali and 6,6 water: then 100 grains.of this falt fhould contain 64,95 acid, 32,29 vol-alkali, and 2,76 water. 123 grains of his aerated vol-alkali which contained 53,75 of fixed air, being fatu- rated with nitrous acid, afforded him in one experiment 127 of nitrated vol-alkali, and in another 123; by my calculation, this quantity of vol-alkali fhould afford 32,6 of the cryftallized. falt. CoRNETTE: [i a8er J Cornette faturated 2304 grains of nitrous acid whofe fp. grav. was to that of water as 10 to 8, that is, 1,250 (he does not mention the temperature) with 1152 of an aerated vol-alkali ex- tracted from fal ammoniac by a fixed alkali (he does not tell how much air it contained), and evaporating to drynefs obtained 1476 of uncryftallized nitrated vol-alkali, Mem. Par. 1783, p. 748. If the fp. grav. of the acid were taken at 60° it would contain by my table 31,62 per cent. real acid, but if at 10° of Reuamur, as is ufual in France, it would contain 32 per cent. the concrete alkali being extracted by a fixed alkali which yields moft, cannot be fuppofed to contain lefs than 52 per cent. of fixed air, and confe- quently 24 per cent. of mere vol-alkali, then 2304 grains of his acid contained 737,28 real nitrous acid, and 1152 of the aerated vol- alkali contained 281,48 of mere vol-alkali; and if 737,28 real ni- trous acid take up 281,48 of mere vol-alkali, roo grains of the acid fhould take up 38,2 nearly of vol-alkali which approaches nearly to my conclufion. Bur as to the quantities of nitrated vol-alkali the difference is far greater; for if 737,28 grains of real acid faturated with vol-alkali afford 1,76 of nitrated vol-alkali, 100 grains of this acid fhould afford 200 of this falt; whereas by my computation it fhould af- ford but 175,44. Tuese difcordant refults evidently fhew that a decompofition takes place in evaporating this falt in a heat even of 809; the hy- drogen ie 3 a | drogen of the vol-alkali partially decompofes the nitrous acid, and converts it either into nitrous air, which by contaét with the at- mofphere reforms nitrous acid, is reabforbed, and attracting more moifture forms the excefs of acid and increafe of weight which is fometimes found; or the acid is fo far decompofed as to become rudimental nitrous air, which is the fubftance Dr. Prieftly calls dephlogifticated nitrous air, which refufing all combination, flies off and occafions a lofs of weight; fometimes both changes take place. BerTHo ter * diftilled 1152 grains of dry nitrated vol-alkali in a hydro-pneumatic apparatus, confifting of a retort, two enfiladed receivers, and a jar to receive air, 1080 grains paffed out of the retort into the receiver, confequently 72 grains only remained in the retort. Tue enfiladed receivers contained 619 grains of a liquor highly acid, and much rudimental nitrous air (what Dr. Prieftly calls de- phlogifticated nitrous air) was produced, the weight of this or other air and water, produced and loft, confequently amounted to 461 grains, for 1080 — 619 = 461. To difcover the contents of the 619 grains of acid liquor he diftilled it in a water bath, there remained in the retort 320 grains Vor. VII. Na of *® Mem. Par. 1785, 316. 28a") of ammoniac, which had not been decompofed by the rft diftilla- tion, but had paffed with the water into the enfiladed receivers, which proves that much of this falt is volatilized during fe eva- poration of its folution. By this 2d diftillation an acid liquor paffed into the receiver, its weight muft have been 619 — 320 = 299 grains, thefe 299 ne he faturated with tartarin, the addition of which produced no {mell of vol-alkali, confequently no undecompofed. vol-alkali remained. He then diftilled cff the water and found it perfely pure, there remained in the retort 54 grains of nitre, whence, depending on Bergman’s calculation, he fuppofes the 299 grains of the acid “li- . guor to have contained 18 grains of real nitrous acid, and that the remainder; viz. 281 grains muft have been water formed ; hence he concludes, 1ft, that 760 grains of nitrated ammoniac were de- compofed, for 72 + 320 = 392 efcaped during decompofition, and thefe being fubftrafted from 1152, leave 760. adly, That from this decompofition 281 grs. of water had been produced, and even more, for fome was loft, p. 318. All thefe changes were effe@ed by the rft diftillation. I sHALL now examine this curious experiment on the grounds of the foregoing theory. 1ft, [ wes Th. tft, 760 grains of nitrated vol-alkali contain, by my account, 57 per cent. nitrous acid, 23 per cent. vol-alkali, and 20 per cent. water. Confequently of acid 43.352 vol-alkali 174,8 water [52,00 760 Acatn, §4 grains nitre contain, by my account, 23,76 real acid, and thefe, fubfiracted from 433,2, leave 409,24 to form water and the rudimental nitrous air. Hence 760 — 23,76 = 736,24 grains form the quantity to be accounted for; we mutt alfo affign the reafon why rudimental ni- trous air, and not mere nitrous air, was left. adly, Of the 281 grains of water, found by Berthollet, 152 pre-exifted by my theory, confequently the formation of 129, and of the additional quantity loft, muft be accounted for. To effec ' this we are to obferve, 3dly, That according to Berthollet’s analyfis roo grains vol-alkali confift of 19,34 of hydrogen, and 80,66 of mephite, confequently that 178,4 grains of vol-alkali contain 33,8 of hydrogen. 4thly, 100 grains water, by the moft exact experiment, rTe- quire for their compofition 14,338 grains of hydrogen, confe- quently 129 grains of water require 18,497 of hydrogen, g Ann. Nn2 ; Chy. [ 280 Chy. p 453; confequently 129 grains water require 18,497 of hy- drogen, confequently there remained 15,3 grains of hydrogen for the formation of about 100 grains more of water, which were loft. sth. Lavoster afligns to 100 grains of fully oxyginated ni- trous acid about 64 of nitrous air and 36 of acidifying oxygen; but in its common ftate of oxygination we may aflign it 25 only of fuperadded oxygen; and confequently roo grains of the com- mon acid contain 75 of nitrous air and 25 of acidifying oxygen. Nitrous air itfelf contains about 3 of its weight of oxygen, and + mephite. 1 Lav. Elem. 235, and Mem. Par. 1781. Now 100 grains water require for their formation 85,662 of oxy- gen, therefore 229 grains of water would require 196,16 ; but 409 grains nitrous acid, fuppofing it even fully oxyginated, contain no more than 147,24 of acidifying oxygen, therefore the remainder v.z. 48,92 mutt have been extracted from the- nitrous air, and much more, if we fuppofe the nitrous acid to contain but 25 per cent. of acidifying oxygen; for then the nitrous acid would fupply but 102,25,and confequently 93,91 fhould be taken from the nitrous air. Now, according to the experiments of Dr. Prieftly and Dieman, if much oxygen be fubftraGted from nitrous air it will be con- verted into rudimental nitrous air; thus this. converfion, and the quantity of water found, are adequately accounted for on the theory above laid down. THE [.285 ] Tue account of the refults of this operation may be rendered ftill clearer by the foll@wing table. 1080 grains paffed into the receivers at the firft diftillation, namely, Of 409 nitrous acid,}} from which its acidify- | ing oxygen, namely, 102,25 graims, were ex- tracted, there remained 306,95 grainsof nitrous air; and of this, after the extraction of 93,91 ef oxygen, there re- | mained 212,84 of rudi- mental nitrous air. J Grs. undecompofed ammoniac - 320, undecompofed nitrous acid - 23,76 water of compofition - 152, water produced - - 229, mephite of the yol-alkali - - J41, rudimental nitrous air - 212,84 Total - - 1078,60 Lofs unaccounted for - 1,4 1080,0 Remarks’ Remarks on Mr. Richter's Calculation of the Rroportion of Ingredients in Neutral Salts. Stnce the publication of my laft paper Mr. Richter, an able German Mathematician and natural philofopher, has publifhed an elaborate treatife on the fame fubje@, in which infinite labour and great mathematical ingenuity is difplayed; his conclufions, however, differ confiderably from imine; leatt this difference among fo many experiments fhould fuggcit a doubt conceining the deter- minations I have endeavoured to eftablif, I feel myfelf obliged to inveftigate the fource of this difference, and to fhew the inaccura- cy of feveral of his fundamental induCtions. Section if. STOCHYOMETRY, 2 THEILE. By his firft experiment, the foundation of fever of his fubfe- quent conclufions, he endeavours to difcover the real quantity of . calcareous earth in chalk, he found 2400 grains of chalk expofed in an earthen veffel to the greateft heat of a wind furnace (how long?) to weigh, when cool, only 1342 grains, therefore 1000 grains of this chalk would weigh 559 grains, and this without far- ther proof he takes to be the true quantity of lime contained in it. On EE Pae7 4 Ow this experiment I remark, that it does not clearly appear that the chalk was thoroughly calcined, but on the contrary there is great reafon to think it was not, becaufe chalk has never been known to contain fo large a proportion of lime as $52., it is true, he fays, it did not effervefce with acids, but furely it heated and bubbled, and fuch bubbles are not diftinguifhable from real effer- vefcence, where the quantity of fixed air is fmall, but by weigh- ing before and after the addition of an acid, which he does not fay he had done. Dr. Brack found it impoffible to calcine any confiderable quan- tity of lime in an earthen crucible, but was obliged to ufe one of black-lead to avoid vitrifaction, 2 Ed. Effays, 219. Smith found the fame difficulty to effeét the entire expulfion of fixed air, Differt. de Aere fixo, p. 40, 43. Chalk in general contains no more than 49 or 5c per.cent. of fixed air, and the chalk he ufed, if it was purified, as he mentions in the 2d feGtion, muft have contained abundance of. moifture ; it commonly contains but 41 per cent. of fixed air, and the proportion of earth in fuch cafe is only 50 per cent. or =3,°;, therefore -£2~ grains of fixed air re- mained unexpelled. Section 3a, 5760 grains of fp. of falt were faturated with 2393 of the afore- mentioned chalk, and the whole being evaporated to drynefs and heated [ 2684 : heated to thin fufion, weighed 2544 grains, now at the rate he had before laid down, the 2393 grains of chalk contained 1337 of lime, and dedu@ting this from the falited mafs, he concludes the remainder, viz. 1207 to have been mere, or what [| call real ma- rine acid*. There the error committed in ftating the quantity of lime is important, as from this the proportion of real acid in the {p. of falt is deduced, and applied in calculating its proportion in other muriatic falts. Ifthe chalk contained 50 per cent. of lime, as I ftate it, then 2393 grains of it contained 1196 of lime, and deduting this from the 2544 of falited lime, the remainder, viz. 1348 is the quantity of real muriatic acid contained in that mafs, and confequently that which is contained in 5760 grains of his fp. of falt, and 1000 grains of it contained 234,03 nearly, inftead of 209, as he ftates it. Sefion 33d. I pass to this fection, as it is here that the defe& of his determi- nation will more clearly appear. In this he tells us, that he fatu- rated 1760 grains of a folution of mild vegetable alkali with 2740 grains of the above mentioned fp. of falt, evaporated and fufed the neutral falt thus formed, and found it to weigh 1856 grains, whence, as by his ftatement, 2740 grains of that fp. of falt con- tained * By an error of the prefs it is ftated in the original that 1207, 2544 : : 1000, 1107. ee ll : : 3] ag J tained 573 of real acid, and this quantity entered into the 1856 grains of neutral fali, it follows that by fubftracting 573 from 1856 the remainder will exhibit the weight of the alkali, namely, 1283 grains. Ir muft be allowed that this is a very indirect and improper me- thod of difcovering the real quantity of mild alkali in the alkaline folution, for it comes loaded with the inaccuracies attending the two previous determinations, that of the real quantity of lime, . from which that of the marine acid is inferred, and that of the marine acid, from which this. laft determination is deduced ; be- fides, if any muriated tartarin exifted in the alkaline folution, as it often does, it would efcape this method and could not be de- tected. But a more apparent objeCtion lies to it; if 1586 grains of mu- riated tartarin contain only 573 of real acid, then 100 grains of this falt would contain only 30,86; now if any thing be well proved in my effay, it is affuredly the affertion, that 100 grains of this falt contain nearly 36 of real acid, being confirmed by the experiments on falited filver, and the decompofition of com- mon falt, therefore Richter’s determination is erroneous, by allow- ‘ing to this falt too {mall a proportion of acid. But if we determine the quantity of alkali in the 5760 grains of alkaline folution by the quantity of real marine acid it was able Vou. VII. Oo to [ 29° J ‘ to faturate, calculated as I mentioned in the above experiment on lime, it will be found very exadtly ; for there I ftated that 5760 grains of his fp. of falt contained 1348 of real acid, and confe- quently 2740 contained 641,25. Now as 36 of acid take up 64 of vegetable alkali, 641,25 take up 1140 of that alkali, and the fum of both v. x. 1781, will be the quantity of muriated tartarin thus formed. It is true he found its weight to be 1856 grains, that is 75 grains more than by my calculation, but this excefs moft probably was caufed by the muriated tartarin previoufly ex- ifting in his alkaline folution. His mode of obtaining what he’ calls a pure alkaline folution renders this highly probable. To obtain a pure alkali (§ 33) he fimply pours cold water on common pot-afh, and leaves them together, frequently agitating them for 24 hours; the folution thus obtained he evaporates to drynefs, and then again treats the faline mafs with cold water, but with a quantity of it too fmall to re-diffolve the whole; fuch was the alkaline folution he employed. Now though much of the neutral falts contained in pot-afh may thus remain undiffolved, yet fome certainly will be taken up, and among the reft muriated tartarin, which is frequently found in vegetable afhes * and does not require above three times its weight of water to diffolve it. ‘To this, then, the excefs of 76 grains may well be afcribed. Tue juftnefs of this conclufion is ftill further confirmed by exa- mining his experiment on vitriolated tartarin. He faturated ano- ther * Wiegleb. uber die Alkalifche Salze 98. 2g 4 ther pound of the alkaline folution with 3647 grains of dilute vi- triolic “acid, and after evaporation and ignition found the falt to weigh 2090 grains, and as he thinks he has proved the quantity of alkali in 5760 grains of the alkaline folution to be 1283 grains, hence he concludes the quantity of acid in the 2090 grains to be 2090 — 1283 = 807 grains; if fo, vitriolated tartarin fhould con- tain but 38,6 grains per cent. of acid, whereas it has been proved But allowing the quantity of alkali in to contain much more. the pound of alkaline folution to be, as I ftated it, 1140 grains, then as 55 parts alkali take up 45 of real vitriolic acid, 1140 will take up 933 of this acid, and the fum of both will be 2073, which differs from 2090 only by 17 grains, owing probably to the mu- riated tartarin contained in his alkaline folution, which may even have been decompofed by the vitriolic acid. He determined, it is | true, the quantity of vitriolic acid by another operation, § 18, but here a material and evident error occurs, as I fhall prefently fhew: rft, To 8460 grains of vitriolic acid, whofe fp. grav. was 1,8553, he added 19200 of water, or, which is the fame thing, to 84,6 of the concentrated acid he added 1g2 of water, and found the fp. grav. of the mixture 1,214. 2dly, He faturated 9075 grains of this dilute acid with 3215 grains of the chalk above-mentioned, and as by his account 1000 parts of that chalk contained 559 of lime, he concluded that Oo 2 3215 [ “agz J 3216 grains of it contained 1596 of lime. Then having heated the felenite thus formed to a degree fufficient to convert lime-ftone into Jime, he found it to weigh 3600 grains, and deducting from this weight that of the lime, he found the remainder, v. z. 2004 grains to be the weight of the vitriolic acid which was con- tained in go75 grains of the dilute acid liquor, and confe- quently that the 3647 grains of it which he had employed in faturating the alkali in the former experiment contained 806 grains. Here, not to repeat with refpect to the chalk what I have al- ready fuggefted, | fhall confine myfelf to a /igle error, becaufe it is manifett : As 1000 parts chalk (he fays) contain 559 of lime, 3215 grains of it fhould contain 1596, whereas by the rule of proportion it fhould be 1797,185; then deducting 1797 from 3600, the re- mainder, v.z. 1803, and not 2004, fhould be the weight of the acid part of the felenite; and 3647 grains of the dilute acid em- ployed in faturating the alkali fhould contain, by his own ac- count, 722, and not 806 grains. It would ill become me to reproach Mr. Richter with this overfight, as many of fuch have often efcaped my notice in my own calculations, and occafioned me infinite labour in rectifying their numerous fpurious confe- quences, TABLE at i es Nate a A [ 293 ] Gk a OOM aie Upc! me | Quantity of Real Acid taken up by mere Alkalis and Earths. 100 Parts. Vitriolic. | Nitrous.| Marine.| Fixed Air. Tartarin - 82,48 84,96 56,3 : os almoft Soda - - 127,68 EG 4T S| 73,41 66,8 Vol-alkali - 383,8 247,82 | 171 Variable Barytes - 50 56 gi,8 282 ae - | 72,41 85, t6 46 4352 Lime - | 143 179,5 _ 84,488 | 81,81 Magnefia - 172,64 210 111,35 as Fourcroy Argill - - | 150.9 335nearly Berg. Be ey eae ae TAB) 4s She ‘Of the Quantity of Alkalis and Earths taken up by roo Parts of Real Vitriolic, Nitrous, Muriatic and | Carbonic Acids, faturated. 100 Parts Tartarin.} Soda. Vol-Alkali. Barytes. Stronthian. | Lime. Magnefia. ce ee | ee | et ee a | —| | — | eS | Vitriolic - - - - 121,48 | 78,32 | 26,05 200, 138, } 7es 57,92 Nitrous - - - = £17,7 73543 40,35 ; 178,12 | 116,86 5557 47,04 Muriatic - - - - 177,6 | 136,2 58,48 314,46 | 216,21 118, 3 89,8 "Carbonic - - - - 9551 149,6 > 2 35455 | 23% 122, 5°, . Toh. 1B A RE IN: Quantity of Neutral Salts afforded by roo Parts of the above-named Acids when faturated with the above-named Bales. 100 Parts \Tartarin. Soda. Alka Barytes. Stronthian. Lime. Magnefia. Vieiotic Ja2t,48| { 145 Geiccared (8294) 309 238 Aer ort te Glauber - P Do, - = > Vitriolated Vol-alkali - = Barofelenite - - 2 Vitriolated-Stronthian - - Selenite - - - Do. - Do, - Dos = “ate Epfom . Do. - - = Allum - - ~ Do. e = Vitriols Of Iron - ~ Do. - - Lead - Copper 5 Zine - - Vor. VII. ae TS SA SB eae. ee; Vi. Of the Proportion of Ingredients in the following Saline Compounds : - - Cryftallized. | eee : - Dry. Water. State. Fully cryftallized. Deficcated. Natural or ignited. ural or ignited. | Naturallif pure, or artificial ignited. 100 Parts Nitrous. itrated Soda = | Do. - 1|Nitrated Vol-alkali Nitrated Barytes - Nitrated Stronthian Nitrated Lime - . °--* 297 Bafis. Acid. Water. Stare. a | a a es 51,8 44, = - | 4,2 of Compofition {Dried at 70*. 40,58 53,2 - ~ | 6,21 of Compofition |Dried at 400°. re | ———— Tenited. 57 2, srr, : - |Cryftallized. 36,21 31,07 7 Zilgngaare - |Cryftallized, Welldried, that is in Air, | Cryfallized, Nitrated Magnefia_ |22, 46, x - py = = |Cryftallized. | Dried at 80°. ae Slt » *+* Tbe cotemporancous 3 Ld z, 2, ry! ieee ~ etx (values of « and z Letalfo a4, @, a, ah &c. | ; pee b, 3, BP Ber be differences of the relpec- c, pM tor € tive orders. TuEN by the theorem for differences. e Spr.= etnep oir Mate i" ¢ +, &c.. where- PLY 2g nis the number of fucceflive values from x to x + x, or from z to z4z. Now if 2 be increafed fine Jimite, any affigned number of: terms of this quantity approaches to the fame number of terms in: the ferjes, n? b paige hte a 1. 2.3. 5 wear x“ +, &c. as its limit, or becaufe z= . te- : b ie a its equal 2 + , —=-, &c. # x % x” * we ‘ Ere b s ord . - Whence the limiting value of — = @ = 2, becaufe & = 2 x ‘ x xe * x ; 3 : 3, c z b PS of — = ~— = —, becaufe —= —-, xs x% x x“ x* &c. &c. Ad, nN. n—I Whence the limiting value of z + #@ + a b+ Sy c + &c. when w is increafed fine limite is 2 + z z ie aaah &c. And becaufe when the former feries terminates its value is z + z+ and when it converges its limit is alfo z+ 2 sw 2¢- FH S+ ae eee + &c. when the feries terminates or Se converges. When it does not converge, nothing can be afferted of it, becaufe we cannot reafon concerning a limit which does not exift. Problems bs gay a Problems for. finding Fluxions per Sa/tum. Pros. t. To find the * fluxion of «" when x does not flow uniformly, and m denotes any whole, frational or negative number. SotuTion. Leta, 5, c, d - - « - = 0, y, B, @ be the i, 24, get lags fey kt ie N—4, N—3, 1—2,2—1 fluxions of x. Tuen the z* fluxion of x” — m—1? mx x + 146 1. NI. Nom 2 b - —a os | P I. 2. ete I. 2 Fi aS fe el) Mh, tl x 2. —1. fe. 71 Sa n.n—I.n—2 PB ks Bt AB Ae, CEM, by SEES Tra Oy I, 3+ 2. FoR einen 2) &e. &e. Tue following are the laws of this feries : 1. ‘THe index of « diminifhes in each term by unity, and is to be continued till it becomes o or m—z. 2. THE coefficient of « is the product of mm—I. = = ‘m—v—t, into the fum of quantities, with numeral coefficients annexed, deduced from the different fluxions of «. 3. THESE [ 328 J] 3- THESE quantities are formed by multiplying together a number v of the feveral fluxions of x, fo that the fum of their exponents pars fhall be x, Thus if abcd be one of thefe quantities pt 2g + 3°r4+4s=nandp+togt+r+s=4. NorrE—By exponent of a fluxion is meant its order. Thus the exponent of d or of the fourth fluxion of » is 4. i pars Cae 4. To the quantity acd - - x is to be annexed, for 2 coefficient, a fraction the numerator-of which is 2. 7—1. 7—2 - a — gq ee k+1, and the denominator pXp—1I. - - IX 2.9.g—I1 - PAs HORE FEEIES patra. 1X8. eT Sa or yee 9. 2 Mesa So ae oT kk—t ~- 1) xo%o—1 - - 1. The lawof continuation of which is evident*. The Demonftration, as far as regards the 1#, 24 and 3° laws of the feries, is readily deduced from confidering the manner, in which the m fucceflive fluxions of « are derived. ‘The demonftration of the fourth law is fomewhat more difficult, but may be deduced as 2g follows: A quantity a 4 prefixed to a power of » is evidently de- rived by taken the ‘fluxion of x, p + g times, and of a, ¢ times in every * Since writing the-above I find that Dr. Waring, at the end of his “ Meditationes Analytic,” {peaking of « methodus deduétionis & reduétionis,” mentions this problem, and gives the three firft laws, in which indeed there is no dilhculty; the fourth, the only one difficult to inveftigate, he does not give, nor does he mention any ufe to which the problem may be applied. [ 3a9-} every different order, with the exception that each 2 muft be taken before the 6, which is derived from it. Confequently the coeffi« cient of sy muft be the number of ‘thefe different orders. This coefficient may therefore be deduced either from the do@rine of permutations or from that of probabilities. The former method is certainly the moft natural, and at firft fight may appear fhorter: but the latter is more readily. ap- plicable to general expreffions. And from it the coefficient of a,a,a - (p) 6, 6,4 - ° (gterms) in the order in which “oom “they are written. The marks underneath fhewing the a: from which the correfponding 4 are derived. The inverfe of the fraGion expreffing this probability is the number of different fg orders, and confequently the coefficient of aé. The prob. that p an a, from which a é is not derived is taken firft is ae that ano- eas . pI ther a of the fame defcription is taken next is —, &c. fo that the probability that all the a' of that defcription are taken previoufly to | p PO sas aa of the as, from whence the é: are derived. is — oaeat. oid i po pee I ; ; ; - - rar That an a is taken next is certainty or m—p—tI Vor. VIL. Tit 29 Eegge | ,, vit £4. The Prob. that another a is taken next is 2 acta oe u—p n—p—1 caufe each a, befides its own chance, has the chance of the 4, which is derived from it, &c. &c. Whence it follows that the probability that all the a@* will be taken before any of the 4 is pA aan Fe Sa c ety ek aX got aX n—l m—2 _ m—p—t n—p 2—p—i SRT KS Oi eae ihe probability « thataeaemne UP h n—p + g—I derived from the firft a is taken next is > &c. n—p + Wuence the prob. that -the whole will be taken in the order in which they are written is q—t Posie dur sekign Lae bal iS G os soiled Sciltou The re- Zs 21 - - = "hated, ciprocal of which fraQion is the seater of 4 ie And by the fame procefs the general coeff. of eS ey as given in the 4% law is readily deducible. Tue dem. by the method of permutations is concifely as follows. If the quantities a, a, a, (p) 4, 6, a, 6 (29) were all different, the number of orders is 7 7—I. - - 1, but as p quantities are the fame, the number muft be reduced by dividing by p.p—t - - 1, or the number of permutations of [ 9324] of p things, and becaufe the permutations of two things are two without regard to order, when the order is fixed the whole number of permutations muft be alfo divided by the number of permutations in each order that is fixed, that is by'2 x 2 X 2 X &c. (7) = 2?,alfo becaufe g a' are the fame, it muft be divided by 9X9—1* - - 1. Whence the number of permutations or the coefficient of a? 87 is as above tated; &c. &c, EXAMPLE I. ‘The 6% fluxion of «” = er i 5 ie <3 6xx ans “ G20 mene Or amet 4 m2 3 m—3 2 m—4 MevwM—2 45 x2 x? Mee. .M—3 x > Smee ISK in moi ised | 3 10 «* 60% xw ihe eee es arh 15 Bh lee —4e +m = = m—5 xo x Exampce II. The 8 fluxion of x? when x —o0, and the * atc and 7, or the fluxions of the uneven orders are alfo =o is ae ¢ a a 56xx”+ 70 «2. For in this cafe the coeff. of x x — FEE unin 4 $. 7. 6. 5 56, and the coeff. of x* = aa 2 = Ow A gm 1 X z1 a * Exampce Il. The coeff. of x* in the 7 fluxion of x3, when “the even fluxions are = o is aut'2 7 6. 6.5.4 1% 7:05 Ny bi ip RE eet 3 6 .3 3 ¢ 6 (8.7.6 = 70K x QIK? WX Oe Jr B. 7.6 2 . 7 id 7 7S nad hae ik Cor. Ifa@=a, b= thy ¢ — —— &c. The denominator 12 Qs , p q r c . oe of the coefficient to be affixed toad bc - - « isp.p—t a ai g.q—l. - Ds Re oe ¢xXo—I - -1 and the numerator Ty eee ey ee, ee PRrogLeM 2. To find the # fluxion of xy z (m quantities.) Sotution. The z!4 fluxion of xyz (m quantities). a a n xyz, &c. + xyz, &e. + xyz, SC. a—t!I nu—I + “ax v2, Sc. + ny 2 &e + &e. m—2 . +a.n—1 2 yz, &c. + &c. To form this quantity the fum of &c. ee AE all the xyz, &c. muft be taken where 2+ 0 4+ 7 4, &. =2. Affixing when @ or @, or y, &c. =o, inftead of x, x, inflead 5 ade of » F gas 4 8 wh aeneeys ofy,y,&c. The coefficient of xyz, &c. is readily deducible by the methods in the former problem, and is = MN—Ii. m2 - = = = 1 Bee yo ST Te EX Boak ate n Ixyxy—I* -~--1x, &. ProsLeM 3. To find the #* fluxion of the fine of an arch x taken m times when the arch does not flow uniformly. Sotutron. Radius being unity. The # fluxion of the fine of mx tS = Mx MR —=1,2 m” 5, nk —, bic. 2. A—1. A—B 4 "3 xx Tue following are the laws of this feries : 1. THE quantities to which the produ€s of the fluxions and their coefficients are affixed are fucceflively mes, mu: m*s, mw: m! cs, mx: &¢. The fign is + or — according as the number of pre- ceding terms of cofines is even or odd. 2. Tue [ 334 J 2. Tur number of fluxional fators tebe affixed to the 7u term | is r, and the fum of their exponents is to be zx. Thus if 2 x? x x? x &c. be one of thefe produdts p+ 29 + &. =~, and ptgt &. =7. : z 3. Tue coefficient of x? , x?, &c. is as ftated in Prob, 1. Exampre. The fourth fluxion of the fine of 3%, whenx =o NOES 27 is 3 X—3.2. x x. Cor. The 7 fluxion of the cofine of mx is had by fubftituting ” Gn the above feries for the cofine of mx, —s, of mx, and fors, . Mk, CS, MR. The Application of the preceding Problems. Prosiem 5. The relation of two quantities being expreffed by one or more equations to find the value of one of them in a feries afcending by integral powers of the other. Sotution. Let # andy besthe two quantities to find x in a feries, afcending by integral powers of y. Compute from the given equations, by help of the preceding problems, the values of x (A), x (B), x (C), &c. when _y = a given valueas a andy =j—a making y flow Ley flow uniformly, Then by Taylor's theorem whilft y changes its value from a toy, x will from A become A + e. + ae x Bie &c. pe Oe sea Aa For more readily ufing the preceding problems, it will ge- nerally be of ufe to clear the given equations from fractions, furds, &c. and fometimes alfo to take the 2d, &c. fluxions generally, in order to’have a more convenient equation, from which the par- ticular fluxions of the higher orders are to be deduced. The parti- cular fluxions of the different orders are to be taken per /altum by’ the preceding problems, fubftituting at the fame time whenever con-- venient the values of x, x, x, &C. _previoufly found. Tue utility and practice of*this method will beft appear by” examples. ExampLe J. From the cubic equation x3 + gx +r = 0, to deduce the values of w in a feries afcending by the powers of 7. Sotution. Let the fucceffive fluxions of this equation be - M 1 &c. and mo taken by Cor. Prob. 1.. making a= «, b= eye tay r conftant. - E $36 J le 38 +P ater me. 243% fp ge 1. 2. ib a2s 3o 2K oe Oe 34.30 +9. 1.2.3643-2-1,40.3-2% 42.3.2 =0, 4%, 3x 49. 1.2.3.42+4. 4. 2 1 ae yo . siiimaed oo ae ee th, QW? b Ge Fe 2s 30 4e $0 KS 4u 3-2 1 ad 4,3 ate 5%. 3 q 3+ 40 56 F 5-403 gant 5% hee ey 54.3.2. 1 be 5-4-3 4 | &a &eo. &e, Hed eas ey SSIS ord@= ake — 30849 ° 3a 4g ? d= Bi2 gen 1 get ad +. be. 3.20% 4 3.a°¢ + ab* . Batt aimee 3x7 49 &c. &c. Cautine the exponents of the quantities a, 4, c, &c. their places in the feries, and the exponents of a p: of rid 2p, &e. the law of continuation is eafily had. For the numerator of the quantity, the. exponent of which is m, confifts of two eens the firftt of which is 3. 2. x into a coeff. which is the fum of the products of every two quantities, the fum of the expo- ponents of which is #, and when m is even, the fquare of the quantity ae ae quantity, the exponent of which is 4 m is to be added. The fecond term is the fum of all the quantities 30° 0, 3.27¢¢, fo that the fum of the exponents of each quantity — m: and when m is a multiple of 3, the cube of the term, the exponent of which is ee is to be added. Now when 7 =0, x} ++9¢x=0, and the values of x are oa, + y—g, fubftituting thefe values in the values of a, 4, c, &cc. found above, and r for 7, we have the three values of 444 +c¢+&c. 2 bes . rn =xt+—Tt &c. the three increments of x, while r from o be- 2 comes 7. Let thefe values be A, B, C, and the values of x are yg By Vg 3G. Tue preceding is given as an example of the method, and not to fhew its fuperiority to others. Since by affuming a feries for x, and making ufe of the multinomial theorem, the fame conclufion will be derived by a procefs equally fhort. Yet it muft be ob- ferved, that the multinemial theorem is only a particular theorem far lefs extenfive indeed in its ufes than the method here given, and not at all more ready in practice. Vou. VII. Uu EXAMPLE , acy ie Exampce II, Given the fine of an arch (A) to find the fine of 2 times that arch, | Sotution. Let » = fine of A, y = fine of x A. Then y S nx Vy Vix fluxions generally making w flow uniformly, and dividing by v. or y, x I—x* = m7? x 1—y?, taking the YX I— eh? 8 e S n? x24, Tue m—2 fluxion of this equation being taken by Prob. 1 and 2, when x = o and y = 0, m m—2 m—2 m—Z I+ m—2.m—3 y Mra —m—2y x? = — mn? gry m m2 ory = me iye2'y Bi Ne : Now becaufe when x andy —0, y=” x and y — 0; it follows therefore that all the even fluxions of y are - 0; and taking for m the odd numbers 3, 5, 7, &c. and x for x, we have y =y ab Bd SS) Bae eae anit fa By — &c a,b, 1.2 Inn Bef &c. being the preceding terms. Alfo if 4 and / be the p—1, hi fea ay 2 p— |? and pterms /=(becaufe m= 2p—1) = —k x rig iy ae THE [#3390 | Tue above folution affords a confpicuous inftance of the advantage of this method, in the ready manner in which the ge- neral law of the feries is derived. This feries has been in- veftigated by feveral authors fince Sir I. Newton, who firft invented it. But all have only deduced a few of the firft. terms, without any proof whatever of the law of the feries. Indeed to have deduced by any of their methods even the 10” term would have been an almoft infuperable labour. Exampte III. To exprefs the hyperbolic logarithmic fecant by a feries afcending by powers of the arch. Sotution. Let a, s, and /be the arc, fecant, and logarithmic fecant, rad. being unity. By the nature of the circle ne Diet and alfo put Sos? ay ao pf = * rot, or taking the fluxions and making 4 conftant, . aii=at*asssaixas xloriaast=h + a? (A). But when a = 0, s=1, «J = 0 and/=a ys'—i =0: whence from the equation A it follows that all the uneven fluxions of Jare =o, becaufe any odd fluxion of the equation muft contain in each term the inferior odd fluxions of 7, For the conveniency Uua2 of [ 340i of applying Prob. 1. let x — /, and (A) / — a> 4 x*, taking the f ae m—2 fluxion of this equation by Prob. 1. / = 2x”x. m—2 4 37-5 u—2. n—3. n—4 5 tt y—2 - - n—6 ae 2X 5 ee Tog NOM —— s Bear hig cae ce ara ah ae fubftituting for the fluxions of x, 4 4, &c. and dividing by I, 2 = - # we get the general equation n bt n—2 4 Z is Z n—2, a ele Sy x == $+ 24 ——— tT --7 I.2 1.2-2—2 m—I.n I--4 n—4 Z n—-A4 Pah? 3 —_ x —— + &e. when — is odd to be continued oS > feat — te 2 n 2 m—2 Mir | / to a hs terms. When Pas even, the laft term 1s —_,— Loe amet mn on Ne Wuence taking 7 = 4, 6, 8, &c. 4 2 4 6 ie Ae Siete ee emma 2) Pe eS DQ -,\Be Oe Bile ED ot Be Bre Te 34355, ye gn 7-B a1 a°° 2———_—. + &c. eet Cy Ae EXAMPLE [) 34t..] Exampte IV. Having given the logarithmic fecant of 45° to the modulus radius (r) = s, and the logarithmic. fecant of any arch a—s-+/ to find a ina feries afcending by the powers of /. Sotution. Let # = the fecant of a. Then by the circle r? : £ alfo becaufe s+ / log. to the modulus 7, /= a— : adn —r @ === or (A) a’ * 2*—7*? Sr? J+, To facilitate the FR 2 Vn*—r . - 3 computations of a, a, a, &c. from the equat. A, when /—o, aa ” a = the arc of 45, and when alfo x* — 277, let a = x Now becaufe 2 = ie : 1 nt : ft. Fluxion of 27 = 222 = = =2r/, fubftituting for Z, 7. al - 20n' x = 23 PR. ad. Fluxion of 7 — — : ee 2 fn m* Fluxion of 2* = WHENGCE [343 ] Wuence from the equation (A) B x #—7? = 7? 7, we have by Prob. 2. 5 R 2 2rJ= B ale (A) Bir? + Be? 77 = oor teary aR P oy i) (A) Bre+ 2B2a,.7/+B23- 7} =o o B= —— ne a , 2 fs 3 (A) Br er 24s Biers Bo want eee 7s 208 —. nie 3 Z eae 25 ft (Ay Batt 4 Biaghed 46a koe 4B ad, Biers 4 Is , or B= 2400 5 4 2 94]3 . 25/4 (A) Br 4 5 Ba PAP GeBowtt LOIN pects Bree at B 6 5 4624 17 24 coor Ba. &c. Now when a = 45°, x =4=/, whence taking the fluxions of the equation x* = B by Prob. 1, and fubftituting for / 2, we get P : 423 “ . 2 f* rape B=" OF Drei ee op. ° atx on » ne 4 3 8 73 oli ae Bo AS ora 2 es if ™ Z Pa eae PO PU eee a Nil ROS 2Ie+2.300=B=— ae feck a aE Dee ae g400/*) 2 F604 Sn al jaea ea nar & 3 2. C*e § Gages 2 i 20 x5 + ty 2B aghee ore ao = = Ee 2.10% # 5 &e. &c Hence while / by flowing from o becomes /, a from a femi- quadrant becomes = ; 3 } i fe a femiquadr. + ae haa ROAR ERE be ASAE os gee Feiss: \Rsal ye 3.7 PAO? Ie ‘Tue two laft examples are feries of Gregory’s from the Comm. Epift. For an account and different methods of inveftigating them fee Scrip. Log. Vol. III. preface and pages 443, &c. 480, &c. Exampie V. To expand the multinomial, a+6z+cz 4 d23 + &c.\ where z is of any denomination whole, negative or fractional. (De Moivre Mifcell. Analytica, p- 87). Let [ 344 ] Ler A. Bi a+6z+ cx +234 &c.) Then making z — 0, and fubftituting for Zz B= I Cate te or BAe 12 B= 1.2.3 dz3 B AR 1.2.3 4 4 B B= 1. 2. 3.4¢e24 oes &c. &c. &e. Wuence from the equation A = Br we have by Cor. Prob. 1. é t—= A=zna_ bz “- td = —, tah A= oo na wee nn—1 a2.) be n—2 bez 42. u—1. n—26' 2° 3 n— A= 9.9 70 “dats + 3.2.4.mu—Ia &ec. &c. “ . A a—t 2. ge nmn—I.n—2 "3 oA=atA+ 7, tk.=a-na éz-+-—" a os aera a b3z3-+&c, I. 2 nu—t m—2Z n. aoc2z*+an—t a bcz3+&e. I | --+na d 2} + &. &e. &c, THE [ 345 ] Tue law of continuation as far as regards the produéts of z, and its coefficients 2, 6,.c, &c. is evident from Prob. 1. and agrees with that given by De Moivre. The law of the coefficients of thefe products is alfo immediately derived. For let 2 pind eve q+2r+3s be any produ@, then becaufe it occursintheterms A, it follows from Taylor's Theorem and Prob. 1. that its coefficient is SS nae n—p eee —_ Ct etait on od a jaan a QXgQ—1xX--1 xX rr—t Lxsxs—IX = _ 2. NI, n—2 - - + n—p—1_ (porg +r -+s terms) The Comes aa tee 9X Q—IK-2e TX rxXr—l xs - IXSXS—HIXK -- = 1 been demonftrated by De Moivre for integral values of 2. 2 4 Exampte VI. From the equation(m) az + bz + “2 4 4% 4 &.= ey + hy* + cy? + ky* + &c. to find the value of z when z and y begin together. Soturion. Taking by Cor. Prob. 1. the fucceffive fluxions when z andy = 0, and » is fubftituted for y. (m) az = gy orz == _ Ay putting A = © (m) as 4. 1.2 A* by* = 1, 2,Ay* on paar: y? = By? Vou. VII. Xx (m) [; 446 "J 3 3 (m) az 4 1.23 9.26 twh Bb eh TQ) Qi Aseys = v9. gay, z 1 2 ABL—A* « c .——— y? = Cy” 1.2.3 a 4 8 (m) az + 1.2.4.3.%1 ACO + 1.2.4.3 Bebyt + 1.2.3.4.3 4 k—-1.2 AC6—B*b—3 A’B BSB = 5,2 R Oe Rec.) Coe: ot = Ley ¢ Sy yee rae ee peo carl saat Seah gam rye ob ip oh. Soy sre de a coefficients of the preceding terms. Tue laws of continuation are readily derived by help of Prob. r. for calling the exponent of a, 1 of 6, 2 &c. and of A, 1 of B; 2 &c. the coefficient of y” is a fraction the denominator of which is a, and the numerator the difference between the coefficient of y” in the given equation, and the fum of products of the capital and fmall letters with numeral coefficients derived by the following laws: 1. To the fmall letter the exponent of which is # are to be affixed m capital letters, fo that the fum of their exponents fhall be m: this is to be done as often as poffible with each fmall letter. 2. THE oe [Say J 2. Tue numeral coefficient of any produc A?’ BC. 2 ce ~-PEGHTXPH Ag+ 37 XP $294 371 8 1.2---- ptog+3rxp.p—i x- ~~ 1X gxg—1xg—2 === 1x71 =a OnE SAIS Bik de ——— =. = the number Beeps ear eg ocy ny oT er Ret I of permutations of AAA (pf things) B B (g) CC (r). Thefe laws of continuation are the fame as ftated by De Moivre*, and deduced by him from the application of the multinomial theorem. Exampte VII. From the mean anomaly of a planet to deduce the eccentric anomaly in a feries afcending by the powers of the excentricity. SovuTion. Let APB be the femi-elliptic orbit defcribed about the focus S and centre C, and P the planet: then drawing RPD perp. to AB meeting the circle defcribed on the diameter A B, the “ ACR will be the eccentric anomaly. Let the mean ano- maly — m (rad. = 1) the eccentric anomaly =c, AC =1, and CS the eccentricity =e. Then m: circumference:: area ASP: area of the ellipfe:: area ASR: area of the circle -.- becaufe CR=1, m=2 area ASR = 2ACR+2CSR=BRXCR+ CSX*DR=cHPes,corm=ctes,e. Xx 2 Ler * Philofophical Tranfactions, Vol. KX. p. 199, Fig. [948] Ler the fucceflive fluxions of this equation be taken by Prob. 2 and 3, when e —o0, andec — m C€41€5,m=0 Craig PE CIGN ME 00 3 C+ 3e¢ C5, M—3eC*5,mM=0 5 ae Sa uest te 3 C+ 4eC3 CS, M—4e. 3ECS, M—AECCCS, M=0 whence fubftituting for e,e €=— CS, MC = — 2CC66S,M = 275,MNX CS, mM = C$, 2m 3 2 C= — 36° CCS, M+ 360275, M= — 3C3XS, 2MxC5,M—3, mM = 3 — 263 x 35,3 m—s, Mm 5 : 3 z 3 C= 4eX Ces, mM— 4065, m—CoS, M= hey 25,4M—S,2M GzGiy 4 cc. a sega ape ee ee = ae Ee ae = 1. 2 oie ti. 4 4 ee oe 35, 3M—S, m4 *25,4m—s, 2m + &c. ‘ vais yi Turs feries is in effe& the fame as the feries ‘given by Keil, but is much better adapted for computation, and befides has the ad- vantage of being applicable to phyfical aftronomy; which the feries of Keil is not.* ‘ EXaMPLE M. De la Grange has given a moft elegant theorem for exprefling in a feries afcending by the powers of ¢ any funétion of x, when x = any fundtion of u-+ 7X, X being a funétion of x. By help of his beautiful theorem, the value of ¢ is immediately deduced from the equation m = c+ es,c. But as the theorem is only adapted to equations of that particular form, it appears equally eligible to deduce the value of c by the above method, becaufe including the demonftration the method of De la Grange is not fhorter. See Coufin’s Aftro. Phyf. Art. 20, page 15. i gaa. I Examp.e VIII. From the mean anomaly of a planet to deduce the true anomaly in a feries afcending by the powers of the eccen- tricity. SoLuTion. Let the femi-axis major AC=1. The eccentri- city CS = e, the anomaly AS T = a, m= the correfponding mean anomaly meafured in the circle the rad. of which = 1, and the periphery P. Then as the areas are proportional to the times, and therefore to the mean anomalies: Fiuvx. area AST : area of the ellipfe: : m:PtST 2 flux. A ey Shoe are eT = Oo Oe SE P “ont xk yI—e, or aX ST*=mV1—e. But by the prop. of the 3 . = : i I—e» ax I—e>| ‘ a ellipfe ST = — . Bence — = pps P eae Memarriy f mor 1—e’' xa Ms Mim ad 4A a feet he i fae A acsa, B= flacs*, a, C= fl acs3,a, &c. and L= ao Then a+2eA+3eBt+4eC+&c.=Lm. From this equation the feries is to be deduced by fucceffively taking its fluxions by Prob. 2. making e flow uniformly &c. e = 0 1.a+2e¢A=Lm=o 2a+2.2e¢A+3.2e¢B=3am Fig. i. See.xd 3 rer Wee ( 3 3.4+3.2c¢A43.3.27Bi4.3.208C—~ Lmao 4 ae mene cee : is : 4, 4+4.20¢A+6.3.2B+4.43.20C+5.4.3.2¢¢D=Lm=45et SECs Secs Now fince A =5,42_ By Prob.3.A=aes,a B=}44+%45,24 A=4¢5,a—a’5,a =fat+facs,2a 3 3, A=aes,a—34a5,4—a305,4 B= Za+4aC5,2a—a25,2a C=3s,a+775,324 and D= 4 @t Zs, 24 Hy. C= Jatsa +4 aes,.3:a &e. &e. Ler thefe values be fubftituted in the above equations, and we deduce making ¢ = e from the 1*, Equat. as—2¢5,m =+£¢75,2m on a 3 eas AUB 2 Pee gic a=—Ox Ys, gmtis, m ; > 4th, a= yx 5,4m+115, 2% &e. &c. e — bb 2e 5 Whence a4 = m + ae hm ti 4 Ne es, 4m + &e, THE [ 35@ J Tue fecond power of the eccentricity or two terms of the feries will be fufficient for the orbits of the Earth and Venus, The third power of the eccentricity or three terms for Jupiter, Saturn, and the Georgium Sidus, and four for Mars. But fix or feven are neceflary for Mercury. It is more tedious than difficult to continue this feries to a greater number of terms. The above folution of this ufeful and celebrated problem, befides being dire is greatly fhorter than any before given: Even than the method of Cagnioli, given by De La Lande, in the third volume of his Aftronomy, edition 1792, where the feries is continued to the ninth power of the eccentricity. Byavery ingenious artifice there given the folution by indeterminate coeffici- ents is very confiderably fhortened. The legality of that artifice might however be juftly doubted, and the truth of the conclufion de- duced fufpected, unlefs verified by other methods. Examp.e IX. From the equation cx we + ye = ay to findy by a feries afcending by the powers of x, 2 being a whole pofitive number (Simpfon’s Fluxions, Vol. I: 293). SoLtution. When wx —o, lety=Y. Then taking the fluxions of the given equation when x = 0, and » flows uniformly. In y x= ay oe 8 3, 24, yx= ay ne”, L oe] n+ a a+t no mN—l--- lex + yea ay sige eee ber, alert ane a) 2535 m+ 1 MBE ee me) we em me a &c. &c Ten becaufe when x =o, and y= Y, y= Y x we immediately deduce z sg ea er Ve 3. Ye cia sed tae co Yat mtr ye E baa oie tata ele a tgs ager cigs tio > ee oe ae ee eee Robie oda +» fubftituting for My 0 sy7~YH+Y < +4 ee a-+1 a+1 n+a 12 oe “2+14 n+i.ntoam Tus example was given to remark that fometimes by this method we may derive a general folution from the particular one. For although the above folution is only a particular one viz. when « is fuch that the feries will converge, yet becaufe we _ eK an 5 to know that 1 + pigs: + &c. = no. the hyp. log. of which is 2 etic n-+2 d alfo becaufe ===— +See t &e. = and allo becaule a+ Ie4 aig FS as + &. = n+4r (e@ X 2-28 Tp apne Te + &.—-14~ = $7" —_- a+ I. a--2t+ 12 or 1.2. --H4, [ a8. 4 x Cary 1.2 ay no. hy. lo were. ——— = n -——-# = ha — - = 1.2. -na" - Y 005: x wv —————,; if M = no. hyp. log. of which is 1, 7— YM 4 12-- Tans --na"? x n a n n— -nmeaXM —1.2 ---mca41.2----nca xn+3.4-°- ma—2 2 n meax4--- - + cxis the general equation of the fluents. : As the above examples have confiderably extended the length of this tract, the fubje@ fhall be concluded by a few obfer- vations. The Theorem of Taylor may be more generally expreffed, for if z be a quantity compofed of two or more independent quantities x, y, v, &c. then while x,y, v, &c. by flowing uniformly become x + Ky J “fly U4 v, &c. z will become z + =~ aiGce. There can be no difficulty in applying what has been before done to cafes of this kind. It may be worthy of remark, however, that by this method when fluxions are fuch that the fluents are expreffed in integral powers, they may be found a priorf: for if z be a function of x, y, &c. where x, y, &c. are independent quantities, and Z the value of z when 4 y, &c. =o, then becaufe z= Z + z+ — + &c. and Pedant —, &c. are derived from z by taking Vout. VII. Noy the eh4 the fluxions, making x,y, &c. conflant, it follows that z may be deduced from z by taking the fucceffive fluxions of z by the former problems. an pate NeoM—T-- - TAFE yn Examp tes. The fluent of x « = Cor. + pu 6 eee The fluent of 3 eyxtwxiy + 2 xy? x + 2«* yy — (taking « and y = 0, and fubftituting for x and y, x and y) 3.3.28 9+ 3. 2x3 7+ 2.3.20 9712.39.20 9° Te 25 3 4 = ey he? yp The fourth example when # is odd is an inftance of finding fluents a prior? by this method. If « = Y_y, where Y is an algebraic funétion of y, then by common algebra reducing this equation to integral values, and taking the fluxions particularly by the former rules, it will be known whether x the fluent can be had in finite terms; in fome cafes, very readily, in many, however, the difficulty will greatly exceed the inverfe method, but this difhculty may be probably obviated by given the fubje€ that attention it feems to deferve. . Bur it ought to be remarked when there are two or more inde- pendent variable quantities, that the given fluxion mutt be poffible, that is, muft have originated from a fluent. Thus for inftance y% is not a poffible fluxion, for it cannot have originated from any flowing quantity wherein x and y are independent. THE oe L eaoete l Tue above method may alfo be applied with confiderable ad- vantage to the finite variations of fpherical triangles, and in many inftances feries may be deduced more convenient in aftronomical computations than the theorems for finite diffe- rences. Yy2 aye PR tee hate ae i aS SMT ET: ae nee i cy. ead bret 0d Tao ene [ 357 ] ACCOUNT OF THE WEATHIER At Londonderry in the Year 1799; ‘By WILLIAM PATERSON, M.D. and M.R.1.A. Months. Es 3 é 5 be | + 3| 8 a gE j}a|s January SW | 10 18 3 February Ww II 14 March WwW 12 18 I April N 15 15 ° May NW} 8 18 5 June N 16 12 2 Tuly Ww 12 15 4 _ Auguft W t | 25 5 September | SE 6 a1 3 /\ in the latter part, it foftened in its rigour, it was often bluftry — March, equally as February, was remarkable for blowing weather, with feveral fqually gales at night and heavy fhowers.—Although the warm winds exceeded the cold in number, in the points taken fingly, yet taken together, the cold were to the warm as 22 to 19; _ and upon the whole it was a-rigorous unpleafant month.— April was remarkably keen and bluftry; the oldeft perfon living did not Peniember fo much fnow in this month ; the greateft part of it fell on the sth, which was little fhort of the 8th of the preceding February with refpe to the degree of wind from the SE. the piercing cold, and drifting {now.—May was alfo cold, with fome fevere blowing weather, particularly the 23d and 24th, which were moft ftormy at night; and the temperature did not foften till the 28th.— June was cold in the beginning, but afterwards contained a good deal of bright, fair, warm weather—During the as) == 31 ° 6 14 ° 28 4 9 9 I Sr pie ae 6 30 ° 2 5 ° 31 2 ° ce) ° 30 2 ° ° I 31 ° ° ° I 31 ° ° fo) ° 30 I o rg ie ’ o field-work might have been better performed by a little attention.— ase yp oe In Oéfober the rain was moftly in frequent heavy fhowers; there were feveral fair intervals; and there were feveral very ufeful frefl breezes. —Hail of an unufual fize fell the 14th, about 3 miles S. E. of Derry.—November was a cold, blowing month, with much denfe fog, and frequent fevere fhowers of both rain and hail; -yet the cold and windy weather, together with feveral fair days and dry intervals, anfwered an excellent purpofe to the farmer——In one of the ftormy nights, the 6th, the large metal vane was blown from the cupola of the Exchange, but no perfon was hurt--Though the prevalent winds in December-were from the cold points, E. and S. E. with fome {mart gales, yet the degree of congelation was net proportionably keen.—Little rain fell; but there was a good deal of foggy and hazy weather. Note.— Tie greatef degree of heat was on the 8th of Fune, when the thermometer rofe to 74°; barometer 30. 31; hygrometer 35% ; wind S. calm, fair, and brigit. barometer 29. 60 ;hysrometer 38 3-4; W. fair, froft, and fog. of 1798, 3 inches, and that of 1797, § inches. Vor. Vii. > ” * The greatef degree of cold took place on the 30th of Fanuary, when the thermometer dropped to are; The annual quantity of rain was about 36 inches, which exceeds that Yy3 vee oy exe eas? ACCOUNT OF TITE WEATHIER At Londonderry in the Year 1799, ‘By WILLIAM PATERSON, M.D. and M.R.1.A. 42 “ ela (vee cS Fa a z 2 St af a |g | 4a 1&6 |e B January Sw 31 ° 6 14 ° February Ww 28 4 9 9 I March Ww 3r 4 4 4 ° April N ° 30 ° 2 5 ° May NW 5 3r 2 ° ° ° June N 2 30 2 ° ° I July Ww 4 3U ° ° ° I Auguft Ww 5 31 ° ° ° ° September | SE 3 30 1 ° ° 1 Ostober sw 6 Br 4 ° 2 4 November | W 6 30 5 ° 6 I December | E I 31 ° 4 13 ° Total - | W | 128 | 198 | 39 | 365 22 25 53 | 9 1798 | W | 126 | 207 | 32 365 22 I4 29 | 5 ty The greater part of the numbers in the roth column denote Lightning alone, particularly in the nonths of September, October, and November, when it took place moffly in the night, and in the ufual nonths for Thunder very little occurred; the caufe of which may be afcribed to the atmofpherical elec- icity having been conveyed to the earth by the conduétor, rain, before it had time to accumulate in the tmofphere and form thunder clouds. es GENERAL REMARKS, Fanuary a good deal of hazy and foggy weather, with both mo- erate and keen froft, whilft the winds were chiefly from the nildeft points; the barometer was many days, at the beginning f the month, above 30, and varied little, though fometimes here was heavy rain; and the ftrongeft freezing took place with he wind at weft.—The winds were in general not only foft in emperature, but moderate in force, there not being more than q or ; blowing days. —Thefe circumftances point out the character of his month as unufual for the feafon of the year; it feems to be emarkable for amixture of gentle winds, fharp froft, and damp, ogyy air—February contained a great proportion of blowing veather, particularly on the 7th and 8that night, when there yere extraordinary high and penetrating fqually gales with confi- ferable quantities of round fhow,; between this and the preceding nonth there were 12 days of uninterrupted freezing; and whilft, n the latter part, it foftened in its rigour, it was often bluftry — March, equally as February, was remarkable for blowing weather, with feveral {qually gales at night and heavy thowers.—Although the warm winds exceeded the coldin number, in the points taken fingly, yet taken together, the cold were to the warm as 22 to 19; and upon the whole it was a rigorous unpleafant month.— April was remarkably keen and bluftry; the oldeft pérfon living did not remember fo much {now in this month ; the greateft part of it fell on the sth, which was litrle fhort of the 8th of the preceding February with refpeét to the degree of wind from the SE. the piercing cold, and drifting {row—May was allo cold, with fome fevere blowing weather, particularly the 23d and 24th, which were moft ftormy at night; and the temperature did not foften ‘ill the 28th.— June was cold in the beginning, but afterwards contained a good deal of bright, fair, warm weather—During the greater part of the fair weather there was a frefh breeze from the. N. ahd fometimes there was a covered fky, threatning rain, though none fell; whilft upon the whole the air poffeffed a confi- derable drying quality.— uly produced a quantity of rain, prin- cipally in heavy fhowers, yet hay was well faved, owing to great abforption and evaporation going on at the fame time, in conjunc- tion with frequent frefh breezes —The leading character of dugu/t was wetnefs; but-as there were feveral frefh breezes and many fair intervals, more might have been done in works of hufbandry than was really effected—The beginning of September was ré- markably warm; and there were fome fair days, though a dufty - like hazinefs of the air indicated much difengaged moifture, which was confirmed by the hygrometer. Yet the nature of the weather was fuch in general, with refpeét to exemption from the moifture colleéting in clouds, and good circulation by the winds, that fieldwork might have been better performed by a little attention.— In Ofober the rain was moftly in frequent heavy fhowers; there were feveral fair intervals; and there were feveral very ufeful fre breezes.—Hail of an unufual fize fell the 14th, about 3 miles S. E. of Derry.—November was a cold, blowing month, with much denfe fog, and frequent fevere fhowers of both rain and hail; yet the cold and windy weather, together with feveral fair days and dry intervals, anfwered an excellent purpofe to the farmer—In one of the ftormy nights, the 6th, the large metal vane was blown from the cupola of the Exchange, but no perfon was hurt--Though the prevalent winds in December were from the cold points, E. and S. E. with fome fimart gales, yet the degree of congelation was net propottionably keen.—Litle rain fell; but there was a good deal of foggy and hazy weather. Nore —TZe greatof degree of heat was on the 81h of Fune, when the thermometer refe to 74°; barometer 30. 31; Mygrometer 35%: wind S. calm, fair, and bright. The greatefl degree of cold took place on the 30th of January, when the thermometer dropped to 21°; barometer 29. 60; hyzrometer 38 3-4; W fair, froft, and fog. The annual quantity of rain suas about 36 inches, which exceests that of 179% 3 inches, and that of 1797, § inches. v Vor. VII. Yy3 " PO nee eer Te Ty. . - 30,29 ott ~~ 2S <* es, = oe baat 25995141 Auguft - ° 30,28 2gt 15 995 is i Pt 10 25285765 } September = = 30,50 | 2¢ : I 7 » “ October - - 30,50 26t ai) E 9753 ae —___ Ei I Br) 1,182292 [aa November 2 S 30,60 lari *3 om 910229 = | ~~ 6, and on 5 fell Snow 1,024653. ; 5 December _ Pa oy 3 30,75. |20t 9 ae ; oS | 7 x60, and on 20 fell fome Snow 22,$84859 Total of the Year. 30,51 - Se ee ee ere 1 . 4 > N. B. The Statements of the Mo/and during tae Months. 2 = = : 4 = - [reso aut January February March April May June July Auguft September Odtober November December | | | | | | | By RICHARD KIRWAN, Ef. LL.D. Pref: R.LA. and F.R.S. Higheft:| Day it happened. 30,60 | 4th, E. ne, 30,38 j2sth, W. 30.46 | 6th, W. & E, 30,50 |14th, E. 30,67 |16th, Var. S. W. & N. 30,61 |rith, E. 30,27 | 6th, W. 39,28 j2oth, W.N. 30,50 | 2d, W. 50,50 |26th, W. |—<—$—$_—_——__.___ 30,60 jarft, W. 39,75. |2o0th, N. to E. 30,51 | BAROMETER. Loweft. Day it happened. 29,38 |23d, NW. to W. 29,10 |2ift, Var. S.to W. & NE. 29,37 |19th, S.to W. 29,05 |18th, E. to W. 29,70 |a1ft, W. 29,32 | 4th, W. 29,35 |18th, E. sth, 29,00 |31ft, SE. & W. 28,86 iit, W. 2 29,276 THERMOMETER. Loweft at | 44, 54,9 Ta 4355 if 5498 | 41, ay 5153 llusase $45 Nicht. Mean, 3 36,75 14,50 36,4 39, 39,9 28, 42575 Synoptical View of the State of the Weather at Dublin in the Year 1799. R Days. 11, and on 2 fell Snow 14, and on 8 fell Snow 17, and on 3 fell Snow 23, and on 2 fell Snow 19 KK T 3554087 1,g18751 3,940975 Gone bk Bae 0,985243 1,123 1 21995041 2285765 1,753733 41,3 13 1,182292 29,47 ad, E. to S. 36,15 6, and on ¢ fell Snow 1,024553 45,06 160, and on 20;fell fome Snow — !22,584859 Total of the Year. x was abfent in England during thofe Months. Yy4 Vex, VI. m, 7 ne ’ ‘. : 3 n " an $ . * i t - A w , , ~ 2 a! . : ; Ss u ; 2 A / oy % z é ee . , ee iin 7 ’ .* / 4 J, Ps ey ers LW sito RIM Ye CH ees, pe Tes ; vs FR tate ERATURE. Dis et fer het tate [wnigeo: fT Some OBSERVATIONS upon the GREEK ACCENTS. By ~ ARTHUR BROWNE, E/q. Senior Fellow of Trinity College, Dublin, and M.R.I. A. a Havi NG lately had an“opportunity of converfing with fome modern Greeks, it appeared to me, that it might not be unacceptable to the Academy to communicate fome obfervations which I made as to their mode of ufing and applying the accents, about the proper meaning and application of which fo much controverfy has arifen. To make thefe obfervations intelligible, I muft briefly recal to the recollection ‘of the Academy fome of the moft celebrated opinions which have been urged concerning thefe accents, both as to their ancient exiftence and as to their ufe. Grevius, Stevens, and Ifaac Voflius in an exprefs treatife on the fubject endeavoured to prove them of modern invention, in- fifting that none are to be found in either infcriptions or manu- {cripts antecedently to the period of about 170 years before Chritt. -Hennin imagines that they were the invention of the Arabians fo December T4th 1799s | 9360 9 fo late as the eighth century, and were ufed only in poetry, and intended to afcertain the pronunciation of the Greeks, and to oppofe the barbarifm of nations who raifed and depreffed the tone of the voice according to the cuftom of their own language without any regard to the true quantity of fyllables.* Wetftein, the learned profeffor of Bafle, in his Differtatio de Accentruum Grecorum antiquitate & ufu, argues for the ufe of accents from the earlieft days, and thinks that when the mode of writing was in capital letters equi-diftant from each other, without diftinction either of words or phrafes, that accents noted ‘by vifible marks were abfolutely neceflary to diftinguifh am- biguous words, and to point out their proper meaning. Tue writers of the laft century were no lefs divided as to the ufe of the accents than as to their antiquity ; fome infifting that they marked tones or intonation—the raifing or lowering of the voice in pronouncing certain fyllables of words; while others confound them with quantity, or at leaft afferted that quantity was in- fluenced or affected by them. Turse difputes have been revived with no fmall ardour in our own times. About 1754, a learned anonymous treatife appeared upon * This feems abfurd, becaufe the accents do not accord with quantity, and therefore would fo have fet them wrong inftead of right. No, the ufe of the accents muft have been to prevent their pronouncing always according to the quantity of the {fyllable, and to fhew them when the Greeks did not do fo. (i: Se 3) upon accents, denying their antiquity and fupported by nu- merous afguments and quotations. About fix years afterwards “Mr. Fofter’s celebrated work appeared, flriving to prove that they were only marks of intonation, and in 1764 was publifhed the Accentus Redivivi of Mr. Primatt, afferting their antiquity, and admitting that they do affect metrical quantity, in fo much, according to his opinion, as to be deftructive of it. From this laft opinion it neceflarily followed, in his opinion, and that of many others, that however it may be right to ufe them in profe, they are not calculated to regulate the recitation of verfe; and hence the common dictum which is fo often heard from the fons of Oxford and Cambridge, that we are to read by accent in profe and by quantity in verfe. Asour ten years fince a {mall work appeared, but of great erudition, fuppofed, and now I believe not denied, to be written by a learned prelate of the Englith church, entitled De Rhythmo Grzcorum; and at a much later period, a Treatife on the Profo- dies of the Greek and Latin Languages, afcribed to another ce- lebrated prelate on the Englifh bench, and fraught with abun- dant learning, and intimate knowledge of Greek literature. In the firft work I would only at prefent refer the reader to the fifth chapter, where the author oppugns the opinion aliam effé in foluta oratione feanfionem rhythmicam, aliam in metris, in oppofition to Vou. VII. Zz Faber, [g62.-)] Faber, Dacier, Pearce, Clarke and others; but from the latter it is neceflary to quote an obfervation or two to prepare us for an application of the facts hereafter to be mentioned. The very learned author, after contending for the antiquity of the accents, totally condemns the rule which has been mentioned, that we are to read by accent in profe and quantity in verfe, obferving truly, that it is not very probable that any people fbould have had iwo pronunciations effentially different, one for profe, and another for verfe. We equally condemns the pofition that profe as well as verfe in Greek muft be read by quantity, that is, as he fays, by the Latin accent, and thinking that the Greek ac- centual marks exprefs the true fpeaking tones of the language, propofes rules of recitation on the bold fuppofition that tone was not always laid on conneéfed words, where the accentual marks appear; whofe pofition however was not changed, to prevent the confufion which would follow from making the pofition of the written mark different in conneéted, from what it is in ifolated words: and he juftly cenfures the printing of books un- accented, one of which, an edition of Theocritus, had efcaped from the Clarendon prefs. He holds that though in placing ac- cent, regard is had to quantity *, euphoniz gratia, and though it therefore may be a fymptom of quantity, itis never a caufe of * For, fays he, the general found of the word will be more or lefs agreeable, according as fyllables at certain diftances from the feat of the acute accent are Jong or fhort. Hence, if accent were placed without any regard to quantity, it would often feduce the fpeaker into a violation of quantity, for the fake of the general euphony of the word. [> eae: of it, and never creates it ; and he calls the opinion of Mr. Primatt and others, that the acute accent lengthens the tone of the fyllable on which it falls, a common prejudice. But he doth not deny. that accent will often be at war with quantity, unlefs tranfpofed in the manner by him recommended. Thus in the line Maya cede Sect Mnryiadew “Ayirn@e, the word, AyiayG, muft be pronounced “AyiAqe. ALTHOUGH I never could affent to a pofition fo ftrongly con- tradictory to the teftimony of my ear as that of the acute ac- cent not lengthening the fyllable upon which it falls; and although my mind was much imprefled with a faying of Mr. Primatt, that it is one of the extraordinary powers of the acute accent, even to change the real quantity, and with his affertion, that the opinion of Meflieur de Port Royal, that the accent only raifes the voice but gives no duration in pro- nouncing, is falfe; I found myfelf difpofed to acquiefce in the fentiment that the accents denoted only tone, or elevation and depreffion of the voice: and this theory feemed to complete the perfection of the Greek language, apparently aiming at more accuracy, and greater freedom from ambiguity than any other language ever did; as to the time of an action by the variety of its tenfes, as to the number of agents by its addition of the dual, as to the object of the act by its three voices, as to the varying pronunciation of its tribes by its analyfis of the dialeéts, and as to the diftin@tion of words written and fpelt in the fame manner, by its accents. We know that fome nations, particularly the Chinefe, have fo ufed the accents. They have, fay : ZZ 2 the fi 38 J the miffionaries, but about three hundred and fifty words in their lJanguage.* Confufion is avoided by the accents, though thefe are not eafily diftinguifhed by an European ear; we knew that this muft fometimes have been the cafe in Greece. as in the inftances of Ava & Aw. ‘The illuftration from our national and provincial accents is obvious 7. Ir occurred to me, however, that it was very furprifing that no author on the fubjet feemed to have taken the pains to enquire what was the pronunciation of the modern Greeks, or their mode of ufing the accents: is it that no inference can be drawn from their ufage, as to that of the ancients? this is eafily faid, but it has not been faid by any of thefe writers. ‘The argument from the Italian pronunciation of Latin giving as no infight into that of the Romans, doth not apply; for the incurfions of barbarous fwarms, like fucceflive over-. flowings of the ocean, have wafhed away every trace of con- nection between the ancient and modern inhabitants of Italy, and perhaps there are more defcendants of the Romans to be found ‘ in * Others fay twelve hundred, and that the nouns are only three hundred and twenty-fix—all monofyllables. From the combination of thefe all their compounds arife. The Greek language has but about three hundred radicals. The Greeks, it has been faid, had but two accents; the acute never rifivug above a fifth higher than the grave, though it might lefs: the Chinefe many,- with intervals much fmaller, and more exactly marked and limited. came do + £.G. a vulgar Scotchman would fay whence you, how you: a common you ou Trifhman, whence came > how do —and an Englifh farmer perhaps would cdame fay whence you. The firft puts the acute accent on the middle word, the fe- eond on the laft, and the drawl of the Englith farmer is marked by the circumflex. Ei wgee: J _ in other countries, for inftance, in Spain, than in Italy itfelf. Certainly the Spanifh language has more obvious affinity to the Latin than the italian has. But the hiftory of Greece has been ‘far diiferent. Twelve hundred years have elapfed fince the Weftern or Latin empire was overturned, but we muft remem- ber that the Eaftern or Greek empire exifted till about 300 years fince, and down as late as the reign of Henry VII; and the’Grecian. people has not been exterminated, buen" remained ever fince, ufing its own religion and language, though in fubjeétion to the Turkifh yoke. It is the fame people, as mich as the Welch are fince they were conquered by Edward the Firft, and I do not fee why their mode of pronunciation fhould be more altered. Twelve hundred years have elapfed fince Latin was a living language, but Greek is a living language to this day. I {peak from my own knowledge when I fay, that the prayer books ufed by the Greek failors, the only defcription of men of that nation whom we ean expe to fee here, are in ancient Greek,—they are able to read the ancient Greek authors, though from want of education not able to tranflate them fluently, and their letters written in modern Greek are eafily to be underftood by us, and differ from ancient Greck, allowing for the ignorance and uncouth ftile of a mariner, little more than one ancient diale@ did from another. I fhall produce one to the Academy, now in my poffeffion. Impressed with thefe fentiments I felt myfelf interefted, when I heard that a Grecian fhip, whofe feizure: has fince been the occafion of a remarkable fuit in the Court of Admiralty, and of the confequent detention of the feamen for a confi- derable Ee joes 4 derable time, had been driven by ftrefs of weather into the port of Dingle in this kingdom. This fhip, called La Madona del Cafo San Speridione, Captain Demetrio Antonio Polo, be- longed to Patrafs, a town fituated not far from the ancient Corinth. The bufinefs of their fuit brought the captain and feveral of the crew to Dublin, and was the occafion of their remaining in this metropolis for a confiderable time. I took the opportunity of frequently converfing with thein, and though their want of erudition and information might feem an argument againft drawing any inference from their practice, to me it ap- peared the contrary, becaufe it gave me the unprejudiced and un- premeditated modes of pronunciation of perfons who could not underftand or know the reafons of my enquiries, or purport of my obfervations. The refult was, to my great furprife, that the practice of the modern Greeks is different from any of the theories contained in the books I have mentioned: it is true they have not two pronunciations for profe and for verfe, and in both they read by accent, and fo far confirm the theory of the learned bifhop, the lateft writer I have mentioned; But they make accent the caufe of quantity; they make it govern and control quantity ; they make the fyllable long on which the acute accent falls, and they allow the acute accent to change the real quantity: in thefe latter refpects therefore they agree with Mr. Primatt, but they defert him when he therefore concludes that poetry is not to be read by accent—they always reading poetry as well as profe by accent. Whether any inference can hence be drawn as to the pro- nunciation of the ancients, I muft leave, after what I have pre- mifed f367,.- | mifed above, to men of more learning, but I think it at leaft fo pro- bable as to make it worth while to communicate to the Academy the inftances which occurred in proof of this affertion more parti- cularly. Of the two firft perfons whom I met, one, the fteward of the fhip, an inhabitant of the ifland of Cephalonia, had had a {chool education: he read Euripides and tranflated fome eafier paffages without much difficulty. By a ftay in this country of near two years he was able to fpeak Englifh very tolerably, as could the captain and feveral of the crew, and almoft all of them {poke Italian fluently. The companion however of the fteward could fpeak only modern Greek, in which I could difcover that he was giving a defcription of the diftrefs in which the fhip had been, and though not able to underftand the context could plainly diftin- guifh many words, fuch as devdpa—Zuroqv, and amongtt the reft the found of AvBeuzos pronounced fhort; this awoke my curiofity, which was ftill more heightened when I obferved that he faid Avépumuv long, with the fame attention to the alteration of the ac- cent with the variety of cafe, which a boy would be taught to pay at a fchool in England *, Watching therefore more clofely, and * Tt will not be fuppofed that this man knew the rule, fi ultima fit longa, acuitur penultima, fi brevis, antepenultima. I cannot avoid here lamenting the total inat- tention to the rules of accent in our fchools in Ireland. Suppofe it to be an ufelefs part of learning, if cuftom in England has made it thought ornamental and neceflary, the Irifh {cholar who is ignorant of it will be cenfured, however undefervedly. I have known men of high literary name in this country who did not know the meaning of the marks which diftinguith encliticks, and gave to oxytones the very converfe of their real meaning. An Englifh fcholar who publifhes a Greek elaflic, could accent it without looking on an accented copy. [gee and afking the other to read fome ancient Greek, I found that they both uniformly pronounced according to accent, without any ‘attention to long or fhort fyllables where accent came in the way ; and on their departure, one of them having bade me good day, by faying Kadnpecgar, to which I anfwered Kaammépz, he with ftrong marks of reprobation fet me right, and repeated KadAyuepx; and with like cenfure did the captain upon another occafion obferve upon my faying Socrates inftead of Socrates. I now felt a vehement wifh to know whether they made the diftinGtion in this refpec&t ufually made between verfe and. profe, but from the little fcholarfhip of the two men with whom I had converfed, from the ignorance of a third whom [| afterwards met, (who however read Lucian with eafe, though he did not feem ever to have heard of the book,) and on account of my imperfect mode of converfing with them all, I had little hopes of fatisfaction on the point, nor was I clear that they perfe@tly knew the difference between verfe and profe. Ar length having met with the commander of the fhip, and his clerk Athanafius Kovouos, and finding that the latter had been a {choolmafter in the Morea, and had here learnt to fpeak Englifh fluently, I put the queftion to them in the prefence of a very learned College friend, and at-another time, to avoid any error, with [ 369: ] with the aid of a gentleman who is perfectly mafter of the Italian language. Both the Greeks repeatedly affured us that verfe as well as profe was read by accent, and not by quantity, and ex-* emplified it by reading feveral lines of Homer, with whofe name they feemed perfely well acquainted. IsHaLu give an inftance or two of their mode of reading: By Oo axttwv mapa Tivee morupnroicboao Jaracons, Tov oO cirroxuerGopsevos mporeon moons wus "AnuAagds, "Es J eperaes emilndes otyeipomev, &¢ 0 exceroeCny They made the « in axéoy— mpocéoy and epéras long. But when they read Kaubé peu, “Apluporoe , 05 Xpuony aupiGeCnnas, They made the fecond fyllable of the firft word Kavi fhort, notwithftanding the acute accent: on my afking why, they de- fired me to look back on the circumflex on the firft fyllable, and faid it thence neceflarily followed, for it is impoflible to pronounce the firft fyllable with the great length which the circumflex de- notes, and not to fhorten the fecond. The teftimony of the {choolmafter might be vitiated, but what could be ftronger than » that of thefe ignorant mariners as to the vulgar common practice of modern Greece, and it is remarkable that this confirms the opi- nion of Bifhop Horfley, that the tones of words in connection are not always the fame with the tones of folitary words, though in thofe of more than one fyllable the accentual marks do not change their po- fition. I muft here add that thefe men confirmed an obfervation of Vou. VII. 3A our WE wie a our late revered and lamented Prefident, that we are much miftaken in our idea of the fuppofed lofty found of woruprcicboo “Saaacoys; that the Borderers on the coaft of the Archipelago take their ideas from the gentle laving of the fhore by a fummer wave, and not from the roaring of a winter ocean, and they accordingly pronounced it Polyphlifveo Thalaffes. I own that the obfervations made by me on the pronunciation of thefe modern Greeks brought a perfe@tly new train of ideas into my mind. I propofe them, with humility, for the confidera- tion of the learned, but they have made a ftrong impreffion upon me, and approached, when compared with other admitted facts, nearly to conviction. In fhort, I am ftrongly inclined to believe, that what the famous treatife fo often mentioned on the profodies of the Greek and Latin languages mentions as the pecu- liarity of the Englifh, that we always prolong the found of the fyllable on which the acute accent falls, is true, and has been true of every nation upon earth. We know it is true of the modern Italians—they read Latin in that refpect juft as we do, and fay, Arma viramque cand, and, In nova fert animus, as much as we. And when we find the modern Greeks following the fame practice, furely we have fome caufe to fuppofe that the ancients did the fame. In the Englifh language, indeed, quantity is not af- fected, becaufe accent and quantity always agree.* Bifhop Horfley endeavoured * The great refemblance between the Perfian and Englith languages, in many refpects, has been obferved by Sir W, Jones.—Here is another: I had the pleafure of hearing oh aed endeavoured to prove that they did fo in Greek, but this is on the bold fuppofition that the accent doth not fall where the mark is placed. , The objection to this hypothefis, which feems to have been admitted by all writers, and confidered as decifive by fome as to profe, by all as to verfe, is that fuch a mode of pronunciation or reading muft deftroy metre, or Rbuthmos. From this pofition, however univerfal, or however it may have been taken for granted, I totally diffent. That it will oppofe the metre or quantity I readily agree, but that it will deftroy the Rhythmos, by which, whatever learned de- fcriptions there may have been of its meaning, I underftand nothing more than the melody or fmooth flowing of the verfes or their harmony if you pleafe, if harmony be properly applied to fuc- ceflive and not fynchronal founds. On the contrary, nothing can be more difagreeable or unmelodious than the reading verfe by quantity, or {canning of it, as it is vulgarly called. Let us try the line fo often quoted— Arma viramqite c4né, Trdje qui primis ab ris, inftead of Arma viramque cand, Troje qui primis ab Oris, or, In nova, &c. No man ever defined Rhuthmos better than Plato, ordinem guendam qui in motibus cernitur; the motion or meafure of the 2A 2 verfe hearing a native of Lucknow, but born of Perfian parents, who was lately in Dublin, AbuTalib Khan, read an ode of Hafiz; accent and quantity always went together: Bedéh Sakée mei Bakée, &c. &c. : with refpect to the pofition of the accent, Sir W, Jones remarks, that the Perfians, like the French, ufually accent the laft fyllable of the word, and the /lrength of accent which he has noted was remarkable in the gentleman Ihave mentioned, and almoft amounted to recitative. [ 372] verfe may be exaét, and yet the order, arrangement and difpofition of the letters and fyllables, fuch as to be grating and unmelodious to the ear. In like manner the feet of the verfe may be exadt, but the ftrefs laid upon particular fyllables of it which follows the quantity may totally deftroy the melody: in fhort, the radical error feems to be the confufion of quantity with melody, and the fuppofition that whatever is at war with quantity and metre muft be at war with melody.* I ardently agree with the praifes of the author of the Accentus Redivivi on the Scholiaftes ad He- pheftionem, that Rhythmus trahit tempora ut vult, & fepe breve tempus facit, ut fit longum; on which the treatife de Rhythmo Grecorum obferves, if this be true, plane a@tum eft de metris. I admit it if they come in oppofition to Rhythmos or melody. With refpe@ to profe I think this is acknowledged, why not with refpe@ to verfe? That it is acknowledged with refpe& to profe, Dacier and Pearce argue from the famous paffage of Longinus, where he fays, that the paflage of Demofthenes fo famous for its pleafing found, rere ro bypicpua, confifts entirely of da€tyl rhythms. ‘Yngicux then as pronounced by him was a dadtyl, not a da@yl meafure, but a daétyl rhythm, and it is re- markable * I {peak with much hefitation, however, when I recolleét, that a moft revered and moft beloved, and truly great man*,who honoured me with his friendfhip, and whofe lofs the world deplores, was of a totally different opinion, and once repeated to me, to oppofe mine, with much emphatis, thefe lines of the third book of the Odyfley :, He rvoc 3° ccviparcey Armrav’ aeginacdrtee Aiwrnyy Odgaror & rorvyarvory iv abavaroros Dativny Kal Sinrcics Bporoicw em} Celdwpor cpegecre * The late Primate. [ass 3 markable that the modern Greeks pronounced it in the fame way; how can it be otherwife if the acute accent be laid on the firft fyllable, Wye, There is a dafyl then in written metre, and a daétyl in pronunciation, and the fame word fhall when written, and when pronounced, be of different meafure. ~ Apply the fame to verfe. ‘Y¥ageue is an Antibacchius for the put- pofe of the poet in meafuring his verfe, but it doth not follow that he may not pronounce it as a daétyl. I dare to fay if Longinus had been fpeaking, not of the mode in which De- mofthenes and all. Grecians pronounced the word, but of the pes of the word, he would not have faid it wasa daétyl. The poet in conftructing his verfe muft take the fyllables as he finds them, and has no power to alter beyond a very little poetic licenfe, for nude conftruction doth not admit of emphafis; but the fpeaker, or the writer are not fo confined, and it was probably to mark their varia- ens to the barbarous nations which overwhelmed Greece that ac= cents were introduced, if they really were introduced at fo late a period. To illuftrate what has been faid, let any man try how eafy it is to make a verfe in perfe&t meafure that fhall be grating or unmu- fical to the ear, and another without mea fure, agreeable and mufical. For inftance, who can difcover mufic in this line, O Fortunati Mercatores, gravis annis, or who would know it was poetry without being told fo. Colitur | 3740] Colitur Hybernia Divis virifque dilecta. is a nonfenfe verfe which has juft occurred to my fancy, in quantity’ perfectly falfe, but in found, perhaps, not unmufical ; and this is the reafon why the Englith have wifely and properly chofen to read Latin verfe by accent and not by quantity, as I verily believe the old Romans did, becaufe they could not bear the found of the verfe when otherwife pronounced ; would the profaic line before mentioned be improved by reading O For tina ti Mér cato rés gravis annis, ? Tue French, though they apply the word accent differently from other nations, may, in my fenfe of the word, illuftrate my meaning; the reafon why the heroic verfe of the French appears fo intolerable to us, is, that we attempt to read it by quantity; it then comes out exadlly like our twelve fyllable verfe, ufually with us confined to ballads, and the famous verfe of Corneille Rome, l’objet unique de mon raffentiment. dances on the ear exactly like Ye belles and ye flirts, and ye pert litile things. But whoever vifits the French theatre will perceive no fuch ridi- culous faltation of meafure, but a folemn and ferious cadence go- verned by accent, adapted to the fubjeét and to the fcene, which almoft prevents the auditors from perceiving that it is verfe. Ir will be here immediately faid, that I confound accent with emphafis: Ido not; [ include in the idea inflection of voice, but in p .876 in a fecondary manner. No perfon can, in my humble opinion, lay a ftrefs or emphafis on any fyllable without making it long, nor is it ever made long (I will not fay it is abfolutely impoflible, I fpeak of the fact) without either elevating or deprefling the voice. Let any man try to exprefs ftrongly the negative, [ cannot, he will fpeak with an acute accent, elevate his voice, lay an em- phafis, and prolong the fyllable. I remember a celebrated member of a houfe of parliament, not long ago, remarkable for his circumflex on this very word. Mr. Primatt highly commends an author on the ac- cents, who fays, no elevation of the voice can be made fenfible in pronouncing, whatever may be done in finging,* without fome ftrefs or paufe, which is always able to make a fhort fyllable long. “I fay, converfely, that no ftrefs or paufe is ever made without fome elevation of the voice, either purely, 1.¢. in an acute tone, or mixed, that is, in an acute tone ending in a grave, and com- monly called a circumflex. Ir will be afked then what is the ufe of metre or meafure in verfe, if we are not to read by it; and here is the grand difh- culty, and I own with candor I cannot anfwer it with perfect fatisfation to my own mind: to thofe indeed who fay we are to read by accent in profe, it may be equally afked what is the 9 / ufe * The treatife on the profodies argues, that in nee length of found and acutenefs ‘of tone are not always united, and endeavours to confute Mr. Primatt, who attempts to account for this, without admitting that it can be fo in fpeaking. . E376. J ufe of long or fhort fyllables in profe, if we are not to attend to them when accent comes in the way: but to gentlemen on the other fide, I'can only anfwer, that in the firft place accent doth not always interfere, and then quantity is our guide, and ac- cent often accords with quantity. Secondly, metre determines the number of feet or meafures in each verfe, and thereby produces a general analogy and harmony through the whole, and it is to be obferved, that, as I apprehend, accent doth not change the number of feet, though it doth the nature or fpecies of them. Thus when we read Arma.virumque cano, Troje qui primus ab oris, we do not make more feet than when we fcan the line, nor employ more time than in pronouncing the next line in which the accent happens to accord with the quantity, viz. Italiam fato profugus, Lavinaque venit. Thirdly, The poet in meafuring his verfe certainly muft be confined to fome certain number and order of long and fhort fyllables, in order to produce a concordance through the whole, and even to regulate the pofition of accent, which though not fubdued by quantity will certainly have fome relation to it, euphonie gratia; but furely the length or fhortnefs of a fyllable cannot determine where emphafis fhall be placed—that muft depend on the meaning and the thouzht; and it would be moft abfurd for the poet to fay to the reader, you fhall not reft upon this emphatic and fignificative word becanfe its fyllables are fhort, and wherever there is a reft, there muft be length and intonation. ad On [ 377 ] On the whole, then, I am inclined to conclude, not only that the ancient Greeks as well as the modern read both verfe and profe by accent, which, indeed, the learned bifhop before alluded to always infifts, but alfo, which he denies, that they fuffered the accents to control and alter the quantity; he does not indeed deny this, if the tones are given where the accentual marks are placed, but he denies that they were fo given. Dacier, Pearce and Clarke admit that they read profe by accent, not by quantity. The learned prelates contend that they could not have had a different modé of reading profe and verfe. I accept both propofitions, though without admitting their inferences,* and the combination of thofe propofitions proves my opinion, which however I do not advance dogmatically or decidedly, but with that feeling which I think becomes every member of this Academy, of wifhing to advance ufeful or ornamental knowledge by free difcuffion and the fuggeftion of fuch ideas as feem to him worthy at leaft of the confideration of the literary world. In the idea that accent muft affe@t quantity I have numerous fupporters as well as opponents. I only differ from the former in thinking that verfe muft ftill be read by accent. I fhall not trouble the fociety further but by the addition of a copy of a letter written by a Greek failor belonging to the thip I have mentioned to the agent fent over here by the Vot. VIL 3B / Turkifh * Of the former that verfe is not to be read by accent: of the latter, that though it is, its quantity is not thereby affected. L a7& ]] Turkifh ambaffador to watch the intereft of the cargo, written in the prefent year, which the latter was fo good as to give to me to fhew the analogy between the modern and ancient language of Greece. It will be obferved that this humble mai:iner ufes the accents with as much attention as any fcholar. Tus letter fo much refembles ancient Greek, that we might almoft fuppofe it was fo, and that the writer had at fchool ac- quired this faculty; but Mr. Barthold, to whom it was addreffed, who perpetually converfed with the failors in modern Greek, affured me that it was entirely modern, and that he could not have correfponded or converfed in ancient Greek. Mr. Barthold had refided a long time in Conftantinople and in the Morea, and was perfectly well acquainted with the language of the modern Greeks. I never faw any book in modern Greek, but I know the New Teftament in that language was publifhed at Oxford in the prefent century, at the time when fome modern Greeks were brought there for education, who, however, by their exceffive idlenefs, difappointed expe@ation. But what fuppofition can be more ftrange than that a parcel of Greek failors, or any one of them, fhould choofe to correfpond in ancient Greek. And I have the pofitive teftimony of Barthold, that this letter is written in the common language of the country, and indeed he defired me to obferve the words introduced from the Italian, fuch as ton intereffon ; and if he had written it from his education at fchool, the termi- nations Dieg| nations and cafes would not be fo entirely foreign from the ancient. I cannot, therefore, doubt, efpecially when | compare it with the language I heard fpoken by all the crew, and when I mention that I faw the log-book of the fhip written in Greek which I could underftand, that this is a fpecimen of modern Greek: the dates and days of the month in the log-book differed from the ancient Greek in the fmalleft circumftance only, thus the oth of January was Iwvwape oydoexarg, inftead of onto xan dexary, I have another of thefe letters in my poffeffion much longer, with which I therefore have not troubled the Academy. I fhall conclude with obferving, that thefe modern Greeks ‘always for accents ufed the word O%ex, thereby confirming the opinion that there is pro- perly no accent but the acute, the grave being the negative of accent ; and we muft remember that the word zgorwdiau, in the an- cient Greek language, is the term ufed for accents: which word, when tranflated into Latin, is accentus or ad cantus, implying _ elevation of voice, or a kind of fong, /uperadded or raifed on the common tone of the voice, and cannot apply to the grave, which is negation of any departure from the ufual level. Tranflation [ 38 | Tranflation of the Greek Letier on the oppofite Page ; * Cork, 1799, Auguft 3d. To the noble, rich merchant, Signior Barthold, humbly, worfhipingly, and lovingly. On the 17th of the laft month I wrote to you a letter from Dingle, writing and exhorting you, that you would take care and better the intereft of me deftitute. That you might know how the other men grieved or held me, often fignifying to me, where againft me they fpoke every day at their mefs, that they would not have me; and IJ again appeafed them, calling and crying out, and to me they gave ear. I exhort you, if you love God, and for the fake of your children, to write me a letter, as how you know ' of your generofity, that I may have and know how I fhall condué myfelf, and that I may convey the men to London, or may carry them to Dublin, and beg that I may have an anfwer how I fhall conduét myfelf, and I fhall as you may direct. 'Thefe, and I remain an outcaft among the mountaineers, Your fervant, CONSTANTINE ANDRIA. * The oppofite is a Fac Simile of the original. : Bee a 4 | lo fare lage 380 fr 8o~ 27 Ig enftst: B— aoe aes Sy 7 tes he yo Xr wacyprlidy oer (O98) Fry way 7 geo Airy Ta fey — = —_— ” 17 9b ae l7- e aoe pap r 1 ors LOADS on? veer Yer Pps dss vIn » per Bork ones was exer egarrtes: ve werte Sos) wy purer 2 iz * te ich bo meer ee fe! % Ov he coe eee te, ae Ovry Ww 2? @sor, — ; Or) 0)? v u- ES Ta ok RCRD a eta > fi wy yor) we pd J ogy oe ain ‘ pope (es - Pee? Seep ants Jaren vd Ak pOtYns “wor papper abs Ios» yoro- ee rs? 2 Sty wsaar > lie Gor X07 ba go oO ISH re SYp par fe Nes argemarin gs ber yikere, ¥ vd spf eo, abby ge be lem gyys fue srepruoes va Ine lel dgouey prors Au va SPvae? 9 ys, os WS pu gleor fs? - = was Kor pis, sea) Ypos7 95 bs pee es ge ph ares, ~ LAx48 ore) rev gavly's spin Lyn wo, yore cutee ye AEA? BOP WAP? 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