¢ . = : , Uj 4 aU) i ? : @ “(ae y » fF i i ‘ _ i ay : 3 iu , ov § Hy - é Le rs re sg » q § in \ f » , ' ° € : rd ' Vih< a . ° Lf t f y fi ad i 4 ® i ‘eet, + i - Ps ' 2 ewe Ky f :~ } ‘ hang d ¥ , ‘i, | my i 6 i : 4 =“ } sOy ad « Pa 7 >» “2 kale Digitized by the Internet Archive in 2010 with funding from University of Toronto http://www.archive.org/details/cyclopaediaoruni19rees ; ek | = 7 P : BrGW O10 PAP s. . EAL Dorttona: : Bp Bvt Mis, ABD Le ATL Oe i, en WORT RS THE CYCLOPADIA; OR, Universal Dictionary or ARTS, SCIENCES, AND LITERATURE. VOL. XIX. THE CYCLOPEDIA; OR, UNIVERSAL DICTIONARY OF Arts, Sctences, and Literature, BY ABRAHAM REES, D.D. F.R.S. F.L.S. S. Amer. Soc. WITH THE ASSISTANCE OF EMINENT PROFESSIONAL GENTLEMEN. ILLUSTRATED WITH NUMEROUS ENGRAVINGS, BY THE MOST DISTINGUISHED ARTISTS. EE IN THIRTY-NINE VOLUMES. VOL. XIX. rr LONDON: Printep ror LONGMAN, HURST, REES, ORME, & BROWN, Parernoster-Row, F.C. AND J. RIVINGTON, A. STRAHAN, PAYNE AND FOSS, SCATCHERD AND LETTERMAN, J- CUTHELL, CLARKE AND SONS, LACKINGTON HUGHES HARDING MAVOR AND JONES, J. AND A. ARCH, CADELL AND DAVIES, S. BAGSTER, J. MAWMAN, JAMES BLACK AND SON, BLACK KINGSBURY PARBURY AND ALLEN, R. SCHOLEY, J. BOOTH, J. BOOKER, SUTTABY EVANCE AND FOX, BALDWIN CRADOCK AND JOY, SHERWOOD NEELY AND JONES, R. SAUNDERS, HURST ROBINSON AND CO,; J. DICKINSON, J. PATERSON, E. WHITESIDE, WILSON AND SONS, AND BRODIE AND DOWDING. 1819. s “502 las ¥. i a: ae a propa ‘ye eeiyl asthenia _ on ho A UY 2U WAVE VEN Aut 5 HoMaerrorrudra'T, AW OTA A ORR! BRR Jere Tet ARuaox noe oxeuaee eae eH =s a ate Anata ae Lene ants, vewirer “se AIX ire ten | = te WOAaAWOS PD he Oe BI a Te OR ay AT wy Ai HR oh GROOT ui" Vie.) Ome ae Ve oh UP) nh? peat, ATA MT + regan wy we ANNO eon th WR : (MIG LiA Bie pele oe ei is svt at, SOK, Oe ele (iA Wrst oar Tear (elo woa Gy A MiNnGts wee o—y ! A ATA FRAY eRe A eae oF Und 7 AC. ets OAD Heir ye hd ce inion 5 yee a ak aa Ye MMT the, CIRCE TA 4 ME Re GVO (OA Bra a ee aoe COE ACHAT ORO 17. RERORD ASI oe; at cro UNIVERSAL DICTIONARY OF ~ ARTS and SCIENCES. INCREMENTS. Calcul des Differences Finies, is a branch of analyfis in- nted by the learned Dr. Brook Taylor, particularly ufeful in the fummation of feries, and applicable to feveral fubje&s of mathematical inveftigation, where fearcely any other me= thod- can be fucéefsfily employed. Montucla obferves, that had the human mind always purfued the path which” Dae ay Mutuon-or, ‘called by the- French ¢ thematicians ; and it was therefore fortunate that any of them would condefcend to illuftrate a fubject in which they could only a& a fecondary part 5; fuch a perfon was however found in M. Nichole, .of the Royal Academy of Sciences, who, having very early been in poffeflion of a copy of Dr, aylor’s work, and perceiving, at once, its general utility, he undertook, the tafk of illuftrating the principles upon appears the moft. natural, the theory of increments, or of which it refted; and, by fimplifying the notation and opera- » finite differences, would have preceded that of fluxions, or the differential calculus; as it feems more natural: for the mind to, be carried from the confideration of finite differ- ences, to that of differences indefinitely fmall, than that the latter fhould be the precurfor of the former. Such, however, was the fact; for the firit diftin& notions of the method of increments did not appear till the year 1715, in a work entitled “* Methodus Incrementorum,’’ &c. by Dr. Brook Taylor, in which both the dire& and inverfe method of increments are treated of in a very learned manner, and an application of the fame to yarouelitere ting problems ; but the novelty of the fubject, and the concile mode of ex- preffion employed by its author, together with the very coms plicated notation, rendered the work nearly unintelligible to any man lefs fkilled in analyfis than the author himfelf ; even the enunciation of fome of the propofitions requires the greateft poffidle attention in order to comprehend their meaning ; but in other refpects, the work bears {trong and evident marks of the lofty genius of its author, and contains, » in the fecond part, many very excellent applications of the preceding theory to the folution of fome of the moft inte- refting and celebrated mathematical problems. Such, how- ever, being the intricacy of the original work, it neceflarily follows that it could only be read by the very firlt rate ma- Vou. XIX. Strahan and Prefton, WNew-Street-Sanai don. tions, rendered it intelligible to readers of an inferior order: his firft paper on the fubje& was publifhed in the Memoirs of the Academy for 1717, which was afterwards followed by two others in 1723 and 1724. Dr. Taylor himfelf alfo,. in the Philofophical 'Tranfations, undertook an explanation of certain parts of his work, and its farther application to fome kinds of feries beyond thofe treated of in the original : the fame was alfo done by M. Montmort, in the Tranfac- tions for the years 1719 and 1720; which latter gentleman, it feems, had conceived fome idea of the theory before Dr. Taylor's work appeared; and a trifling altercation took place between them as to the originality of fome of the no- tions which was claimed by both parties. Tn 1763, Emer- fon publifhed his « Method of Increments ;’’ a work which, at leaft, does this author as much credit as any that he ever produced ; but the notation refembles, in a great meafure, that of Dr. Taylor, which renders it now almott obfolete. Emerfon appears’ to have been extremely anxious to bring the theory of increments te perfe&tion ; and earneitly urged thofe who were qualified for the tafk, to purfue the paths he had opened to them. I cannot,”’ fays he, “ pro- mife that I fhall have time and leifure hereafter to profe- cute this fubje& any farther. And as I have an earnett de- fire of advancing truth and improving f{cience, let me here B entreat i INCREMENTS. ventreat the friendly mathematicians, who are lovers of {feience, to lend their kind affiftance for the advancement of this uncultivated branch of knowledge, yet in its infancy, or rather, as yet, in the hands of Lucina, either according to the model I have here laid before them, or fome better if it can be found, fo that, by degrees, it may at length be brought to perfection.”? Hence it appears that this author was, in 1763, fenfible of the negle€&t that had been paid to the theory of increments ; and even now, near half a cen- tury after this date, the fubject has not been advanced, nay {carcely touched upon by any Englifh mathematician, while foreign authors are filling quarto volumes with the theory of differences finies, and its almoft univerfal application to the mott curious and important of mathematical inquiries. Inu- ler, whofe univerfal genius led him to the inveftigation of every fubje& that was ufeful and interefting, did not leave the theory of increments untouched, but has treated of it, in his ufual matterly flyle, ina work entitled « Inftitutiones Calculi Differentiales,’” &c. in which he has given a new form, and much extended the bounds of this important branch of analyfis: and fubfequent authors have adopted his ideas, and rendered permanent the form he gave it. Vari- ous other works have fince appeared to illuftrate and render familiar the principles of this doGtrine ; the moft complete of which is the * Traité des Differences,’’ &c. par Lacroix. Boffut has likewife a chapter on this fubject in his “ T'raités des Calcul Differentiel,’? where the theory is treated in a very elementary and comprehenfive manner ; and the fame is alfo done by Coufin, in chapter 3. of the introdudtion to his «« Traités de Calcul Differentiel,’” &c. Having thus given a brief fketch of the hiftory and pro- grefs of the method of increments, we fhall now endeavour to explain the principles and application of it; in order to which, that we may prepare the reader for the more general theory, it will be ufeful to confider the fame in a more limited form in the firft initance, in doing which we cannot have a better model than that of M. Nichole above-mentioned. The method of this author is extremely fimple, but cer- tainly lefs general than that of Dr. Taylor; it is, however, well calculated for conveying the firft clear and conneéted ideas of this theory, and prepares the reader for more general refearches, by leading him on, from ftep to flep, with order and precifion. If we confider x as any variable quantity, which is con- tinually increafed by a conftant quantity a, fo that it beeomes fucceflively x, 2 +, x + 2n, &c.: and if y be any func- tion of x, made up of factors, as y= wv (x + 2) (v + 22) (a + 37), then the difference between this value of y, and that which it becomes when w is again increafed by a, is the increment of y, or of x (w+) (a + 2n) (x + 3n)3 which increment is readily obtained, by obferving, that (x +2) ee eT eee ee (vtn) Qet2n) — «x(x4+4+n) = 2n(x +2) (x +n) (w+ 22) (2 + gn) — a (@ 4 1) (x + 2) = 30 (x+12) (x + 22); and hence generally, the incre- ment of w(v +n) (v+ 2n)...(#-+rn2)=(rt+i1) 2 (2 +2) (x+2n).. (@ + rn): Andhence again con- verfely, the integral of (r +1) a (a+n)... (x + 22) (x +rn)=n(e+n) (+ 22)..(4% + rn); 0r more generally, the integral of w (# +n) (vw + 22)... (« + rn) =: Again, the increment ef #” is the difference between x» and (# F n)™. ‘Now (x 4 Bye — 2™ = ma” 'a + m(m—1) (m— 2) Pee 3 by means of this general expreffion, for the increment of xm, we readily deduce the increment of any funétion of ay as am tase" +b, + &c. Thus, for example, if it were required to find the increment of «? + 3x*-+ 4x, we have increment of x? =3jx*n+3xn* + n3 of gee = - O6xn + 3n* —— of 4x = ? 4a Hence the increment of #3? + g¥? + 4% = 34° + w(3n74+6n)+n'+3n% 44m. It is therefore not neceffary, when we proceed thus, that em 2 n+ w"-3n3 +°&c.3 and — the propofed fanétion fhould be made up of fa¢tors in arith- | metical progreffion, as in the former rule. } : She . Zn Again, the increment — is the difference between — and 7 x I n ' ; Ee ree a) and in the fame way we finger ; 1 2n ‘ increment.of ——.__ _—_—__. — ee ee eee x («# + n) x(x +n) (x +22)’ I —3n ~ 2 (e4n) (@ +22) @+ 3a) and generally the increment of aw (« = n) (% + 22) nG io) Eto eee Bh Se ee eae a(x +.nj(e + on)... (etn Ia), again, converfely, the integral of —(r+ij)a And hence, ao (ev +n) (a +22), . Ghee . - . (ee Fee . 1 Fe eae a) (2 Fain) 2s Ghee ne nee a . _ J generally, the integral of x(@ +”) @ +20) (#4 Pn) I — rn ~ ¢ (a tn) (w+ 22) 2. ++. es (ir — In). This, as was before obferved, is not the moft general mode of confidering the fubjeét, but we are much mittaken if it be not the moft obvious and natural; and therefore the beft adapted for illuftration, and for eonveying to a beginner the firft ideas of the theory. We will now fhew the applica- tion of the above principles to an example or two, by way of elucidation, and then proceed to a more general and ex- tended inveltigation of the method of increments, and its application to mathematical problems. ; . Ex. t.—Let it be propofed to find the fum of 100 terms of the feries ’ Oe er ek? Se 100. IOI. Each of thefe terms is of the form w (a# + 1), and it is obvious that the next term to 100. 101, that is 101. 102, is the increment of the feries; or, making 100 = 2, the lat term is x (x -+ 1), and the fucceeding one is (x +- 1) (# +2); which is evidently the increment of the feries, or the @if- ference between the propofed feries in the firlt cafe, and what it becomes when w is increafed by the common dif. ference 1; and therefore converfely, the integral of this in- crement, that is, of (v + 1) ( + 2), will be the fum of H the ae Me™ ‘ethy a 5 INCREMENTS. the feries fought. Now the integral of (x + 1) (++ 2) — *(* +1) (« + 2) 3 100, we have ;-and fince x in the prefent ex- 100 X IOL X 102 3 ample = 343400 for the fum required. Ex. 2.—Required the fum of 2 terms of the feries 1.2-34+2-3-443-4-.5 + &. n(n +1) (2 + 2)- Here, by writing # inftead of n + 1, we fhall have for the fucceeding term x (w + 1) (# + 2), which is the in- crement of the feries, and therefore the integral of Py es ee et) (> ee soit be the fum required; which, by re-eftablifhing the value of n(n+ 1) (2 + 2) @ +3) _ 4 x = n + I, becomes the fum of n terms. Ex. 3.—Required the fum of a terms of the natural feries of fquares 17+ 27>4+ 3° + 4° + .--.. 0m. Here writing x for 2, the fucceeding term is (w + 1)* =e + 2e47+4+1=4(¢+ 1) + w+ 1), which is the increment ; and it confifts of two parts. Now the (w+ 1) (e+1), 3 integral of x (v +1) = 3 d I integralof x» +1 = = fora 8 And therefore fince « = #, we have a(n + 1) + 2 @ oe == the fum required.. Ex.—Let it now be propofed to find the fum of the infinite as I I I I feries —— + —-— + —— + — + &e. I .2 Ar 3 3 . 4 4 . Ge Here it will be neceflary for us to confider the feries as generated from the extreme term; which is o, and therefore I ; : ate 3 —— asits laft term, which will therefore be the increment ai i I + —— + —— + &c.; and, confe- eR Bh 4S : quently, the integral of this will be the fum of the {eries, of the feries : I wanting only the term ace Make, therefore, # = 2, or I i 4 ‘ ; I ieee =e ——~ 3.in which cafe the increment —— be- a3 Ye {at F) 1.2 I . P I comes ———--—., and the integral of this = — (becaufe (#—1)a x the increment is — 1); therefore the fum of the feries, be- ~ ae re , ; A ginning at the term a BF » to which adding the firft . 3 term -, we have the fum of the whole feries = 1. This will ferve to explain the method purfued by M. Ni- ehole in his firft paper, and will be ufeful as an introduction to what follows: in which we fhall not limit ourfelves to con- fidering the continual increafe of x as conflant as is done above; but as being variable like » itfelf; for it is under this form, that the theory of increments becomes fo univerfallp applicable to almoft every {pecies of mathematical invettiga- tion. Notation and Definitions. 1. As the increment of a variable quantity x is nothing: more than the difference between that quantity in its firit ftate, and what it becomes after a certain increafe, this dif- ference may be properly reprefented by Dx, or A x; and in the fame manner, if y be any funétion of a variable quan- tity, Dy, or Ay, will reprefent the increment of y. And as in the fluxional or differential calculus, fis made the cha- racter of integration, fo in the prefent inftance, we fhall em- ploy it to reprefent the integral of any increment. 2. The increment of a variable quantity being, as we have obferved above, only the excefs of this quantity in one ftate, over the fame quantity in the confecutive ftate, it follows that if a variable magnitude x become fucceflively 2,2’, x", - x", &e. we fhall have Aw = x’ —x3; Ax! = x"! — «x'; LD xf a! — es A xl = xl? — x!" & ee It may happen that an increment may be pofitive, or ne- gative, according as the variable quantity of which itis the increment is augmented or diminifhed, with regard to fome other magnitude or magnitudes which we fuppofe to in- creafe, and of which the increments are therefore neceflarily pofitive. ; 3- The increments of quantities being themfelves quan- tities, if they be variable, we may take the increments of them, thefe are called /econd increments ; and if thefe fecond increments be alfo variable, we may in like manner take the increments of thefe allo; which are called ¢hird increments, and fo on as long as the differences or increments are variable. In‘all thefe cafes, the condition of the increments. being variable, is neceflary ; becaufe if, in any cafe, they become conitant, then it is obvious that the increments are ©, whether it be the firft, fecond, third, &c. increment that thus become conftant or invariable. Thus the feries of {quares 1, 4, 9. 16, 25, &c. is an ex- ample of a cafe in which the fecond differences or increments are conftant: for this feries may be confidered as generated by a variable quantity x*; which is fuch, that if the difference between the fucceffive terms be taken, they will form a feries of quantities in arithmetical progreffion, and confequently the differences of thefe differences, or the fecond increment of x” will be conftant. Inthe fame manner we find the third differences of the feries of cubes, 1,8, 27, 64, 125, or the third increment of w, is conftant; and therefore the fourth increment = 0; for after any order of increments ecomes conttant, all the ulterior orders mutt, neceffarily, become zero. 4, As‘ Ax denotes the firft increment of any variable quantity #; fo A?x, Aix, Ata, Kc. will reprefent the fe- cond, third, fourth, &c. increments.of the fame quantity x ; which expreffions are fufliciently diftin& from Aa*, Aa’, Aw', &c. which repegient the powers of thofe increments ; amd if it be required to exprefs.the power of any increment palt the firft, as for inftance the ath pewer of the fecond, third, &c. increment of 2, that will be done thus, A’ 2’, AD oh Shes 5- In fome problems it is neceffary to confider a certain order of the increments as conilant ; thus, in any arith- ‘metical progreffion the firft increments are conitant. In the feries of natural fquares, the fecond increments are neceflarily conftant; as are alfo the third increments of cubes, &c. as we have feen above. But there is an in- B2 finite. INCREMENTS. finite number of quellions, in which, from their nature, it is not neceflary that any order of their increments fhould be con{lant: yet as we may_attribute to a certain quantity whatever variation we pleafe, providing that the variations of the other quantities depending upon the firit be fuch as to accord with the variation we have attributed to it; it follows, that in any problem, we may at pleafure make any order of increments of a quantity chofen at will be con- ftant ; obferving only, that the other quantities ought to vary in confequence; and, therefore, we cannot make an- other order of increments alfo conftant, unlefs, from the na- ture of the problem, fome other increment has a certain ratio to that which is fo affumed. 6. The whole of the method of increments confifts of two problems; viz. 1it. Finding the increments of all or- ders of any variable quantity, raifed to any power; the produ& of different variable quantities ; and generally of any*function of variable quantities ; which problem is al- ways folvible, and prefents but little difficulty in any cafe ; and this is called the Dired Method of Increments. ‘The other problem, which is the reverfe of the preceding, is that of finding the integral of any given increment, which is frequently infolvible ; at leaft, without infinite feries, or fome other mode of approximation ; and this is termed the Inver fe Method of Increments : which two problems we will confider under their diftinG heads. Of the Dired Method of Increments. 7. Since the increment of a variable quantity is the dif- ference between the fums in any two confecutive flates, it is obvious in general, that in order to find the increment of any funétion of variable quantities, we muft fuppefe, that each of thofe quantities is increafed or diminifhed by their refpective increments; and fubflitute thefe quantities, thus changed, into the propofed funétion ; and from this refult, if there be fubtra¢ted the original expreflion, the remainder will be the increment fought. Ex. 1.—Find the firft increment of the fum x +y + 2. Thefe quantities, augmented by their refpeCtive increments, become ° (wtAx) + +Ay) + (2+ Az) from which fubtracting the original expreffion, there re- mains A x + Ay + 42, as is evident; fince the whole increment mutt neceflarily be equal to the fum of each par- ticular one. In the fame manner we find the increment of + +y — x, orA (x+y —2%) =Awt Ay— Ax. And if we had to find the increment of a+ x+y— 2, we fhould have, confidering a as aconftant quantity, A(at+ «x+y —2%) =Axt+ Ay— Az, the fame refult as before, becaufe the conftant quantity a has no increment, or its increment is equal to zero. Hence it appears, that if to the fum of any variable quantity we add or fubtra& any conftant quantity what- ever, the increment of the whole fun@ion will {till be the fame. Ex. 2. Find the firft increment of #”. This is, from what is obferved above, the difference be- tween (a + Ax)” and #3 which by the binomial theo- m(m —1 ) Et rem becomes m x” Aw + 4 oZ w-2 Ag? + $ fer 1 ma) a" Ax + &e. Ax”. Bin Byard Thus we find the increment of 2°, or = 2 mA ae A. at A (x3) = 2m eee a Ae , A (at) = 49 Ae + 6x? Ax? + 4xAx? + of &e. &e. &c. &c. And if the propofed quantity, of which the increment is required, be aw”, a being a conftant multiplier, then it is obvious that Aakers a(x+Aw)™—a(x")= a [Cae ah A i —— wy a aN ce) whence the increment a2” is equal to a times, the incres ment of +”. a(w + Ax)" — ant = Lx. 3. To find the firft increment of the produ& x ys Here w« becomes wx + Aw and y becomes y + Ay whence the produt = wy +yAw+uaAy+ AvAy; from which, fubtracting the original quantity #y, we have A(ey) =yAuw+udy + Ardy. And in the fame way we find (ayes reds azay a wsyAztevAyAe +yArAx+2ArAgy4+AuAyaAg, And in like manner may the increment of any other pro- dué& be readily afcertained. If the product was avy, axyz, &c., a being a cons ftant quantity, we fhould have A(axy) =aAd(xy), A(axyz) =aA(eye), Ke. that is, we muft find the increment as above, and multiply the refult by the conftant faCtor a. Ex. 4.—To find the firft increment of any quantity of the form x (x + a) (# + 24) (% + 3 G)swe(e + 7a). It is obvious that this may be referred to the preceding ex- ample, by making x + a=u; w+ 2a=y3; «+34a=23 &e. under which fubftitution, the function, of which the increment is required, reduces to xu yz, &c. and, confe- quently, A (xuyx) may be found as above. But if, without this fubilitution, we find the a€tual produét, it is obvious that it will take the following form (where A, B, C, D, &c. reprefent conftant quantities) ; viz. et A gt * 4 Bat? + Cz"— 2 Dae ime. and hence by finding the increments of each of thofe terms by example 2, the fum of them will be the increment of the funétion propofed. Thus, A (x (a + a) ) = A: (x?-- a we Now A (4) %= 22Aw + Aa; A (a x) ee A Gx) whence A (* (a + a) ) =(2¥+ a)Axv+ Ax’. And in the fame manner A (x (x +a) (x + 2a)) =A (# + 34x" + 2@°s). Now A (2?) = 32 Ax t+ 34Aa + Ax’ A (342) =6axrA#+.3aAx Aaa ag) = 2a The fum of which particular increments will be the incre- ment of the original fun€tion propofed: and in exaétly the fame way, the increment of any fimilar funétion may be afcertained. ¥ ; Ex. 5.—To find the firf increment of the fra&tion : . Here aks “% INCREMENTS. Here we have etAv wv yAx—why _ 4(=)= y+hy yy VF +¥AY » Vg heey) (th HED * And therefore from the expanfion of (y* + yAy)~* into a feries, we have « (ySa—xAy) ( Ay Ay’ Ay? ) A Ee oe a po) 4 ao ae @ a i 77 ge And if the propofed fraGtion be effected with any conftant fa&tor a, the whole of the above increment muft be multi- plied by a. Ex. 6.—To find the firft increment of the quantity a/ (a + 2°). Here it is obvious, on the fame principles, that A ( /a@ + x)°) = J(@ + («+ Aw) ) — (a + x’) ai (a + 4°) + (24 Ax + Axx’) ) — J/(a@ +. x) And if now we confider thefe two expreffions as two bino- mials, to be raifed to the power 4, it is obvious that from the developement of the firft, there will be cancelled the firlt term, and the other terms will reprefent the increment re- quired ; and thus we have oo 2 ate 2 25 2(@+8)? 8@ +x) Ms in A) 5 16 (a? + a*)” Ex. 7.—Uaving given the equation y? — ax + x7 = 0 which expreffes the relation between the conftant quantity a, and the two variable quantities x and y; to find the equa- tion which ought to exprefs the relation between a, and the firft increments of x and y. Here we mui fubftitute x + Aw for x; and y + Ay for y; which gives (y + 49)? —a (ex + Ax) + (x + Axl =o; from which, fubtra€ting the original equation, there re- mains 2yAy—aAx+24rAx+ Axr’+ Ay=o, which is the equation required. And in a fimilar manner the firft increments of any quan- tities whatever may be afcertained, as alfo of any algebraical equation. When it is required to find the fecond, third, &c. incre- ments of any propofed funétion, it is only neceffary to confi- der the preceding order of increments as variable quantities ; and we fhall thus pafs from the firlt incr2ments to the fecond, from the fecond to the third, from the third to the fourth, &c. ; in the fame manner as we pafs from the ori- ginal funtion to the firft increment. Ex. 1.--'To find the fecond increment of x. Here we have, in the firft place, A (a?) = 2x Aw + Ao the firft increment. And if now, in this expreffion, we fubftitute v + Aw for x, and Aw + A*w for Aw, we have A? (2?) =A (2Ke Aw + Ax*)= 2 (#@+A-z) (Ax + A’x) + (Aw + A*x)* — (24 Ax + Az’) =2he+ 4A Meo + ae hrt Ay, which is the fecond increment required. And in the fame way we find the third, fourth, &c. in- crements of a quantity or funtion, by fubftituting A* x 4+ A'winitead of A’; and A?y + Ax inftead of Aba; aud foon. | &c, 2xAxv+Ax? (2xAxv- Ax’)? Remark.—If we confider the fecond increments as being conttant, it adds very much to the fimplicity of the operation ; for, after having found the firft increments, viz. A (#) =28Ax + Ax’, n(x?) = 3a? Aw be 3 Are + Ax, A (@*) = 4a An F677 Ax? 4+ 47 A254 Ast, &e. &c. &e. &c. we fhall have for the higher order of increments, A? (x*) = 2Ax*, A? x? =o, At (x3) = ©, OC» LA? (ee OR A 2? + CARL’ (27716 A 2, aS (w*) Os ZN («') == On eCee A* (at) == 12-0 A x” + 2447 As bora x, At (a) = 242 A 2+ 2S eee 24 42%, A? (a) = 0; Ue (Ge opie. &e. &c. &e. Scholium.—1t is eafy to find, on the fame principles, the fecond increments of all forts of fun€tions. For example, to find the fecond increment of the product wy, without fup- pofing any increment as conftant, we muit firft find the firft increment, which is, A(ey)=yOu+ueAyt+tAxdAy. And fubftituting now in this expreffion, x + Aw for v, andy + Ay fory; Aw + Mx for Ax, and Ay + A’y for Ay, it becomes (y +Ay) (Aw + Ax) + (4% + Ax) (Ay + Ay) + Ax + Ax) (Ay + 4’y); from which, fubtraéting the firfl increment, there remains A (xy) =syXat+ xy + 2AxdAy +2Ayha+ 2Aurd’y + Ax d’y. And if we fuppofe A. conftant, this expreflion reduces te Mey) = ae hMy + 2ArAy +2Ax dA’y. And fimilar methods apply in all cafes ; it will, therefore, be unneceflary to give any farther examples, except in the cafe of an exponential expreflion, which is fomewhat dif- ferent. Let it be propofed, for example, to find the increment of the hyperbolic logarithm of «. Let »y = 4.1.4; then, asx becomes x + Ax; fowilly become y + Ay, that is, yt Ay=bl (x + Ar); and hence, fincey = 4./. 4, we have gtAy—y=h.l(x+ Ax) —b.1.%, or Ae Bie $As)—ble=bi (1 + =*). Now, by the well-known logarithmic feries, we have Aw A: Ax’ Ax’ A 2+ bi(xy ft) St Oe, Se Oa x x 2a 3 x 4x Ax Ax Ax A at whence Ay = A (4/x) = — — -+—- : x 2x 3x 4 ‘+ &c. as required. And the higher order of increments of 4/x will be found as above, by taking the increments of the terms of this feries. Ex. 2—Required the firft increment of the exponential expreffion a‘. Make y = a’; then, when » becomes x + Aa, y will become y + Ay; whence 7 ot ay = avtar y —— fg a® x a4*; but a‘ whence Ay = a x ad* — a® = a (a4* — 1), 3 Now, INCREMENTS. Now, expanding (a4* — 1) into a feries, we have Ax . Ax . . fqael yyy SR PEER I 1.2 Ais? (la)3 eee. Hye Therefore Ay, or A (a*) = 2 a 7 3 3 a’ x eee Als es xe} I 1s 2 ere as required. And the higher order of its increments may be found by the ufual method. Of the Inverfe Method of Increments. 8. In the inverfe method of increments, the queftion is to find the integral, or funétion, from its increment being given. We mutt, therefore, examine with attention the fteps by which we defcend from a variable quantity to its increment ; and then, by the reverfe operation, we may afcend to the integral, -when the increment is known. But this reverfe operation is attended with the very fame difficulties as the in- verfe method of fluxions, for, as in that, every fluent may be readily put into fluxions, fo may the increment of any func- tion be readily obtained; but it is frequently difficult, and fometimes impoffible to find the fluent of a given fluxion ; and fo in the method of increments, there are many cafes that will not admit of integration; we fhall, however, give fome of the moft ufual and obvious rules, and which will. apply to the generality of examples. Let us firft attend to the powers of a variable quan- tity x. 1. Since Ax = A (wx); therefore, reciprocally, [ax =x. Andif we fuppofe A x as conftant (a fuppofition that has place in all that follows), we fhall have fa aX T=, x Aw 2. Since A (2°) = 2”Ax% + Ax’, therefore, recipro- eally, { (2* Ax + Ax’) = 23 or, which is the fame, rede + fas orAx {1 = 43 therefore /'1 ° “4 7 a = #°; whence again alfo ex + Ax x ; ie ie —— = —~ ; andhence, by tranfpofition, / «= 2 2A x Z DA’ Ar x Ax ee x = ——- = —— — — x I=—— — -. 3 2 2Ax% ee 2Ane 2 3. Again, fince A(z )= 36 Aw+ 3 4A x +Az2'; therefore, reciprocally, {(3 Aa + 3xAu + 4x’) = x*5 or, which is the fame, {3 ct Aw +43 tnx stefan = «7; or dividing by 3 Ax 32 x edafep =" x Ox] x — J sf 3 ;. whence, again, % RAD Miah ge Te fe = — & Md Kak ef 1;0r, which is the fame, Sale - 3 a we! x? xAw on ,, Zax 2 6 We find in a fimilar manner, by continuing to fuppofe Ax as conftant, and fubftitutiag always for the quantities con- tained under the particular fum their refpeCtive values ; the following refults for the integrals of the fucceflive powers of x; in which we have repeated the two preceding ones, for the fake of uniformity. i . Ax fa x” hig * -[9' See ee x3 en xan Ax? = _-— x 34x 2 7 6 , wt x wAQx Angee —-—-+ ; J 4A” 2 4 ; PR g # xt Aw we Ax Mi dient <0 SiN ogee 3 Cee fa : ae a Kgl Nagai aes A x —-— ++——— — ; i . 6A 2 12 ons &c. &c. ; &e. &e. where it is only neceffary to obferve, that if the propofed increment have any conftant multiplier, the integral above found muft have the fame. Cor. 1.—Hence we may find the integral of any fundétion compofed of the powers of x, affected with any conftant co- efficients a, 6, c, &c. For in order to find the integral of fuch an increment, it is only neceflary to find thofe of the ay powers of x, and their fum will be the integral re- quired. Ex. 1.— Required the integral of the increment a + b 2% + cx’; confidering A » as conttant. ae axa fr= — AF ‘c Lae : b x? b: presbfe= ie el PAIN EY 2, 3 2 A ies c fx = (ey SE EA 3A 2 6 And hence by addition, ax bx ot Cm CRE { (a ba +ecu*) = — = aa | Pee ) Rar ae 2 34x 2 * cx Ax Ex. 2—Required the integral of ax! — 64°, confidering A «x as conttant. Here we have fo a x ay ax Aw ax Nz a — — — —-— : Aw 2) 3 30 f pe jige bx? bxAdx * ; img 34x 22 6 the fum of which expreffions will be the whole increment fought. Cor. 2.—When it is required to find the integral of a quantity of any of the following forms, (Aw being fup- pofed conftant,) viz. a (« + a) («+ a) (x + 2a) (2 + a) (w+ 2a) (x +32) (w+ a@) (w+ 24) (w+ 3a) (x + 4a) we arrive at them by taking the actual produ& of thofe quantities, and finding Gieeelevely the increments of each of the termas: thus, « nb f@tays fet fas ae 2 ge Aw 2. f (w+ 4) (w+ 2a) =f[@ + 34e+2a)= x” Pi INCREMENTS. f gal e.g # - gAg) 2 aks _3ax 3ax Ssanno, Z 3 2 Mea ae) ea 2s eae a 3 Ax 6 Zan jax 28x thn. 2 Ax 3- Again, f (x + a) (« + 2a) (wn +34) = S@ +642? + 1144+ 6a')= te. ue a «hes aa eear fax a 6ax? — 6ax GaxAx : 5 gAx 2 @ a Tl aie OO 60x 3 J6a ny we gg’ An 2a Zax (# . a — — 7 ae Ax 2 4 Ax I axAx lias 6a64 Ii1a@« 6a ae ¢ —— ‘ = —— ;and fo ono I 2An Aw 2 Bx? other fimilar quantities. And if it be required to find the integral of quantities of the form x («+ a) x (x +a) (« +22) x (x a) (x + 2a) (w+ 3a) ceesece. Site we have in the fame way x We. ae Are "og a): = is = —- . 1 fix (w+ a) Ss + fax 3Ax ie Et Rta 2h x. 2 2 Sx (#+4) (+ 2a)= [¥' + 3a fx ++ 2a? fx = C A Ra | Se OF BD i Fae 4 4 1 — ee eee * < 3af x Ts 3 Aw \d 2 ‘7 “ > ASS Bae, aN 2 the fum of which will be the increment required. And, in a fimilar manner, we may find the integral to any other quantity of thefe forms. ». Remart.—Before we proceed any further on this fub- jet, it will be proper to attend to the corre¢tion of any in- tegral, when from the nature of the problem under confider- “ast becomes neceflary. As the increments of any variable quantities x, and w + a> are bothexprefled by A.v, the conftant part a having no in- crement, fo, reciprocally, the integral of the increment A x may be x, or x + a; therefore, when we have found the integral of any increment, we mutt add to it aconflant quan- tity, which will be zero, if che integral needs no correction but a real quantity, pofitive or negative, in other cafes, which muft be determined from the nature of the pro- blem. This remark will be of confiderable importance in what follows. 10. Let us then confider thofe produéts, of which the fators are continually increafed by aconftant difference ; Vix. v(x + Ax) x(x + Ax) (4+ 2Axz) a (#+4+Ax) (#+2Ax) (x9 +3Axn) &ce. &e. &e. Here it is obvious that the increment of # (x + Ax); or TA (# w+ Ax)) = + Ax) + 2Ax) — # (x + Ax) 3 whence, by multiplication and fubtration, we have A (x (@ + Axr)) = 2Ax(x+A~x). In the fame manner we find, A (* (@ + Aw) (a+ 2Aix))= 3Ax(x 4+ Aad (x + 2Ax); And thus again, A(#(@+ A#)@+2Aa) (4+ 3Ax))=4de (x + Ax) @W+2Ax) («+ 3 Ax) and fo on of other fimilar products. Whence it appears, that in order to find the increment of any produ& of the above form, we muft fupprefs the firft faGtor x, and write in its place the increment A «, effeCted with a co-efficient equal to the total number of factors ; all the other part of the expreffion remaining as before. Whence again, converfely, the integral to any increment of this form, will be found from the reverfe operation. Thus for example, fz Aw (4+ Ax) = 2(e+ Ax) 2. f3Ax («# + Ax) (# + 2A) («+ 2Ax) 3. f 4Ae (x + Ax) (x + 2Ax) (9 +3 Ax) = x(x + A x) (« +2 Ax) (x + 3A x) and fo on of others. Whence, in order to find the correfponding integral to any increment of the above form, which may be reprefented ge- nerally by ahae(xt Ax) (e+ 2Az)..5 (x +n Ar), we mut change A x in the firft fa€tor into x, and divide the whole by the number of fa¢tors ; that is, faN«(x + Ax) (2 4a Awe. . (x + 2nAzx)lv= ‘ ax (x+ Ax) (a+ 2Anx).... (4+ nAv). n+ ii This part of the theory has been before cenfidered in the preceding pages, in deferibing the method employed by M. Nichole of the Academy of Sciences, publifhed in 1717. 11. Let us now confider thofe fraétions, the denominators of which are eompofed of factors fimilar te thofe abeve de- {cribed, viz. I x (vw + Ax) 1 ——$—$—$— x (¥ + Ax) a(x +x) (w+ 242) (x + 3 Aa); Ke. in which A » is conftant, Here, INCREMENTS. Here, by taking the differences as before, we fhall have Aft te tt -aa ; e(e@+AaS 7 (w+Ax)(e4+242) «(w+ Ar) = — 2Aw x : x (x + Ax) (« + 2A x)” And in the fame manner we find the increment, or, —3A+ . 4| ape eian se I wv (vw + Aw) (wx + 244) (ew + 3Ax) I 3° sie +Ax) (x+2Ax) (x + —=x.)t ehh ; wv (xe + Ax) (w+ 24.2) (x + 342) (w+ 4 Ax) end fo on of others. Whence it follows, that in order to determine the increment of any expreffions of the above form, we mutt.increafe the denominator by one factor, and multiply the new fraction by the conftant increment taken negatively and affected by fuch a co-efficient, as is equal to the number of factors in the denominator of the refulting fraction. And hence, again, converfely, in order to find the integral correfponding to any increment of the form aNtsw pilGieepaieeessmeee se ae eee x(«# +x) (w+ 2NH#) (x +ZAx).. {x +2A.x) we mutt fupprefs the laft fator in the denominator, and afterwards divide the refulting fraction, taken negatively, by the produ& of A x, into the number of factors comprifed in the denominator of the faid fraétion ; thus, an wv Sa Ax) («+ 2Ax) (w+ 3Ax).. (w+nAx) — 2 4 AX X ~ ne (wtAx) (@ HP eAx)(@+3Ax)...(e+a—1AZz) 12. On the fame principles we find A d y= : (os x+(a+1)Ae and, therefore, reciprocally, t I 1) Awa ior x+nAx ~ peace J eit The integration of thefe two latter expreffions cannot be effected feparately, but the difference of them is evidently I “+nAx I equal to the algebraical fraCtion ; x+aAde Whence it appears, that quantities may fometimes admit of integration, by being decompofed into many parts ; which, though they will not admit of it in their ftate of feparation, may, notwithftanding, be fo combined with each other, that the final refult fhall be algebraical ; a circumftance that com- monly happens in prattical operations. _ Examples of Integration. 13. Ex. 1.—Find the integral of the increment 32+ 2A a(x + Ax) (x +242) : Sd This quantity admits of the following decompofition ¢ 3ur+2Au ner 1 bi ae a(x +Ax) (7+ 20x)” Awe(x) ' Da (a+ Ax} 2 r I 1 T ee LR gue et Ax (# + 2ARpe ae pale er ar 2 I — ——— -__——__ : each of which parts isevidently of x + BA a ak the form that has been invelligated in the preceding para= graph. By making firftz = 0, n= 1, andz = 2; thus we have eh Ban I ie ge eo+Arx hie by which means the integral of the propofed quantity takes the following form: wiz. Znr§. = 2 He I i I Axl tye Le Re iy hy A aaa But the above formula gives alfo : I I id — I gf vs Aw Ya am e+Axw x ) Lx, 2.—Required the integral of the quantity 3 ea, xo (% tug x) 3 Ax nt hee “2 (x+3Ax) ks and, confequently, the integral of the quantity will be Mea.) Gia v YT] age eas Now the above formula gives ZO. oud Pe 8 x-+ Aw I we I y. voll SF ff —— 3 of sre ae 2 Am J +246 whence the whole integral is exprefled by 1 ; I I q uA St gk ne 2. eee, =: Sar ie ihe x x + 2%. I —2 ax (ooo — an we. -Awe as required. 3%—Azn x Aw («+ Ar) ° Here we have ey w+3An Bj xe NES But Ps ee a Mah. i. : x -Ax x+2A¢ xe +A by the fame formula, and, confequently, we have at length 3 4% mae KS I coe | J «(t@$ 3 Oe) x+2Ax wtAxn x Such are the elements of the direét and inverfe method of increments, and which will be found to embrace ae number of cafes ; but thofe who with for a more complete de- velopement of the principles, cannot confult a work better calculated to convey the neceffary information, than the third volume of the “ Traité du Calcul Differentiel, &e.”? par La Croix. We thall now conclude this article, by INCREMENTS. by fhewing the application of the foregoing theory a few examples. Application of the Method of Increments. _ 14- The fummation of feries, by the inverfe method of increments, is founded generally on this principle ; that if we have any feries of quantities, as Ly d, e, &C. which are decived from each other according to fome known law; each term may be confidered as the increment of the {um of all thofe which precede it. Thus, let at+b+e+dte=x and a+b+ec+d+e4f=2!: then it is obvious, that 2! — z= Az = f; and therefore, converfely, the integral of any one of thofe terms, con- fidered as an increment, will reprefent the fum of all the preceding part of the feries. This being premifed, we may proceed to the folution of the following examples. Ex, I> Required the fum of » terms of the natural feries Bt+2+35+44+5+--..2 Here, by writing x inftead of 2, the term next in order will be « + 1, which being the increment of the feries, we fhall have / (x + 1) = the fum required. Now f (« + 1) =f * +f; and by art. 8. | as Cy a we d ee coe) as gd ATL lie he 2Ax a z Aw And fince in this cafe A « = 1, we have 2 atx Wan I aE aS Zi! 2 by writing again x inftead of », which is the fum of 2 terms of the propofed feries, as is alfo evident from other conti- derations. Remark. ~This example offers an eafy illuftration of what has been obferved at art.g, of the correétion of an inte- gral, which is neceflary in many cafes, the fame as the cor- reGtion of a fluent is in the fluxional or differential cal- culus. Suppofe for example, that inftead of the preceding feries beginning at unity, it had commenced from any other term, as 7; the general law of formation would have been the fame, andthe increment would {till have had the form x 4/13 and confequently the integral, in the firs inftance, would be reprefented as above ; VIZ. fay === ' 2 “But here a correétion of the integral is neceflary, for from the nature of the feries, when x = 7, the fum of the feries is 7, this being the term at which the feries com- mences ; whereas, without a correction, we fhould have the fum = 28; we mult, therefore, write ' “af x : , | (e+1) = xa +c; ¢ being the correction, and fince when x= 7) a +96 = 75 OF = +. == Se we find c = — 213 after which the fim of any number of aii of the propofed feries is readily obtained. ‘Thus for example, let n = 16, then the {um of the f{eries beginning with the term 7 becomes 16° + 16 — 21 = 115 as required. ; 2 Vor. XIX. Ex. 2.—Find the fum of the feries of f, > 3 +5 + 7 + &e, 2’. -couhisall Here the general term of the feries is (x )? + 2)*, and there. fore the fum required will be expreffed by the Sate geal of (~ + 2)* r Now / («+ 2)*%= f+ + f4 x4 [43 and by article 8 hb gelie ar, ime — 4". os t Jaa SS 3 a 4 fia As x Be ig Fi J@+2~=F- Sing tot 4 an # : ; 3 x 7 2 => + — a= a fe & by making « = 2. But here again, if the feries do not begin at unity, it will require a correétion, which will be found generally, thus : fuppofe the feries to commence at any term f; and iets =," then the above formula gives the fum of the {feries rabies 3 2 term p = Ft - whereas it ought to be 9°; the correction therefore becomes — (& f PRD >) a 2% a 3 ¢ “i _f2ee 7 f) = ¢; and, therefore, th ae ee > and, re, the general expref- fion for any number of terms of this feries, between the limits p and a, will be nn G2. § ae gas Cor. —This example will alfo furnifh the folution of the pro- blem, when the roots of the {quares differ from each other by any conftant quantity m; for it will only be neceffary to make Ax = m, and the fame formula will give S (x +my= S x + am J «+ mf fez x ee. aS {AR 2 6 2m? 2m 2am jJx=—— — 2 Ax 2 m x m fs =— Ax And fince in this cafe we have A x = m, the fum of thefe expreflions becomes -— wee . - + rs + ¢, the correétion ; and therefore the general formula for the f we ie 7 term py. will Ar or the fum beginning at any fte+mt ae 3m . oy n omy Pp Pf mp Ste eo \ ge | a GT : Ex. 3—Required the fum of the natural feries of cubes rm+2+ 353+4'4 &e. a’. . C Here INCREMENTS. Here the general term is (2 4-1)*; and the integral of this, that is, [ (x a%: r)5 = [x' + 3 fx + 3 fix + fu is computed as follows : } vt aw wAw { 3 — Cl aaa - —— . 4Ax 2 4 why! 2 a sg? gL 2 3 ] pies ipedh eA ity. Stes : . 3 Ax 2 6 rp ace q-, a — -— 3 / 2A & 2 . x J I= - . Ax And here, fince Aw = 1, we have x x x x? + ay” e+33ao— + --4—- = ( ) be ( ) 4 2 4 2) : Rae ab aa \. Ot, by making x = 2, the fum required is onan Hence a very curious property with regard to the fums of confecutive cubes beginning at 13; viz. that this {um is equal to the {quare of the fum of all their roots. Thiasss 13-23. (1.4.2)? 4 234+ 33= (1+ 2+ 3)’ P+ 2+ 3h4 4s (1+24+ 3+ 4)" &c. &e. As to the correétion in thofe cafes that required it, it will be found as in the preceding example ; as alfo the fums of any number of cubes whofe roots are in arithmetical pro- greffion, having any common difference m. And the fums of any powers whatever are attainable upon the fame prin- ciples. Ex. 4.—Required the fum of any propofed number of terms in the feries of triangular numbers I, 3, 6, 10, 15, &c. In this cafe the general term is AR 2 and it will therefore be neceflary to find the integral of cases) (eas) af (x +3) («+ 2) but (art, 10) 2 =f (e+ (+2) = “(rapa alter wd ae by making # =, and Xv = 1. 5 The fame may be otherwife found from art, 8; thus, 4 of (et) @ +2) = Ffe tase aa [2 x xv vw AL PCRs Eg S IZ oat 4 ee piper esi Aah And fince A x = 1, this fum becomes. a as | @ tm Gerd) = ae ae + == 3 ae edaiet 2 gy And the fums of any order of poly- gonal and figurate numbers may be found in the fame manner. vie. Ex, 5.—Required the fum of any number a terms of the feries 1.2+2.34+3-44+ 4-54 &c.n (n+ 1) Here the general term of the feries is x (x + 1), it is therefore required to find the integral of (« + 1) (# +2). Firft by art. 10. we have : yr (# + 1) (aw +2)= any I) (at a2 iti ) ie ) by writing x = 2, and Aw =r. & (el Ly Rae aye 3,42 ae The fame may alfo be found, as in the example above, by article 8; but it is needlefs to repeat it, as it only dif- fers from that in its conftant factor. ° We have in neither of the foregoing examples any correction, for it is obvious that when w = o, the formula x(x +t 1) (x 4 2 eg) = 0, as it ought to be, and there- fore the integral needs no corre€tion. Ex. 6.—Required the fum of any number of terms 2, of the feries 1-4..7.10 + 4.97. 10.13 +7. 10. 13 .16 4b) &e. a(z +3) (# +6) (n + 9). Here the general term of the feries is (v7 + 3) (w + 6} (v + 9), and we have therefore to find the integral of (w + 3) (w -& 6) (« + 9) (x + 12). Now, by arts ro. JS (2 +3) (2 +°6) (#49) ©€+ Bes ( «(x + 3) (x + 6) (w© + 9) (ew + 12) 5 Ax n(n + 3) (2 + 6) (a+ 9) (2 + 12) SY by writing x =n, and Aw = 3. - Ex. 7.—Required the fum of the infinite feries i I I I meat AHA aa ga A ime Fg eh aa Sige 5c paige dae 1 ————__—__———.._ Here the general term of the feries a(n -+ 24) (n + 2) s is : x (wv + 1) (a + 2) to confider the feries as generated from its extremity, which is infinitely diftant ; under which circumftance each term will be, the increment of all thofe which follow it in the above arrangement ; and therefore, in order to find the I x (# + 1) (x + 2) fimply to find the integral of ; and the mott fimple idea is now, fum beginning at any term » we have I Cre ea taken negatively, becaufe. the increment «, that is, Aw=-— 1. Now, by art. 11, I Be a IP SETS (a +P) "OR # 2% (a fry 1 becaufe A 7 = ~— 1; that is, the fum of the 20(4 + 1) above infinite feries beginning at any term s Be n(a + 1) (a+ 2 is equal to ; and therefore when » = 1, the I 2n(n + 1) whole feries becomes = 3, Gor. %) Moe C t Cor. Having thus found the fum of the whole feries, we may -readily find the fum of as many terms as we pleafe, for we have feen that the {um beginning at the nth term from the firfl, is ; and therefore, beginning at the oy 2n(n + 1) I . ; 2(n +1) (x + 2)’ which laft fum taken from the whole fum 2, will leave that of the firit 2 terms, that is, Leth F RUE Eek. Saat a(e+1)(e+2) 4(a+1)(2 42) Thus, if x = 1, fum = 4; if » = 2, fum = aie if nize 2, thé! fum = %, &e. Ex, 8.—Find the fum of the infinite feries I I I prep ey 8G. 13 ite aD Here the general term of the feries being x (2 + 1)th term, the fum will be = fumz terms = + — TT TR.17 I —___—_——; we (7 + 4) fhall, on the fame principles as in the precedimg example, have to find the integral of Gua. Z Now by art. £1. — i! I ED iat KS = —;becaufe Aw = — 4. | es RR as Making, therefore, » = 1, we have } for the fum of the whole feries, beginning at the firit term. We might have added here a variety of other examples, had the limits of the article admitted of it, but fuch as are given will be found to apply in a number of cafes; and will indioate to the in- telligeut reader the method of application in many others, which are reducible to fimilar principles. It will be proper, however, before we difmifs this fubject entirely, to fay a few words on the celebrated theorem of Dr. Vaylor, the learned author of this theory. ‘This theorem may be exprefled as follows : ' Let Y'reprefent any function whatever of the variable quan- tity ~ ; then if w be increafed by any difference A x, the value Y, that is Y + AY, becomes (adopting the difleren- AegdY AxadyY tial notation) Mee deYo = Net a re Awd Y A xt Bh gy or AY = C24* r.2.3da° 1.2.3-4dx Teda# Ageaik Oe a' yt Aatd*Y ‘haa oe 3 Ine 2a oe 4ube* Jaw of continuation is obvious, and requires no farther de- velopement; it mult, however, be remarked, that when A « is negative, the terms of the above feries mutt be taken plus and minus alternately ; and it will therefore be more ge- nerally expreffed by AwdY ide 31 ae In order then to find the increment of any funétion of a variable quantity, we mutt take the fucceflive order of its fluxions, by which means all the fluxional parts in both the numerator and denominator will difappear; and we {hall have the value of A Y expreffed in terms of a and A x; and this expreffion will always be finite, unlefs the function be tranfcendental. Let us propofe the fun@ion Y = ax + ba* + x to find the value of Y', when x becomes vw + Ax, + &c.where the $AY = so bibacsk I. 3ae* — INC d.Y =adu+2zbrdx+ 32°dx e2y= 2bdx* 4 6xdx* — ay= 6 dx} d*Y= fo) Now thefe values being fubftituted in the general feries, we fhall have for the increment of Y, AY=Ar (at2bx+ 347) 4+ Axe (b4+ 3x) + Ax Again, given Y = ax + bx? — cx} + x, to find the increment of Y. dY=adx+2bxdxr~— 3cex dv+4a'dx GN y= +2bdx —6exrdx’ 4+ 122dx CY = — 6cdx? + 2444.33 aX + 24d2t aY= ° Whence making thefe fubflitutions in the general theorem, we have axe gays abx— 3cx* + 4x) + Ax’ (b— 3¢x + 6x") + Awe (c+47) 4+ Ax. Thefe examples will explain the method of applying the theorem to any cafe that may arife, it being as univerfal ia its application in the theory of increments, or differences ; as the binomial theorem is in the expanfion of roots and powers; but, like the latter, it was left by its author with- out demonftration : it has, however, received many fince from feveral able mathematicians, though there is no one of them perhaps that is quite fo fatisfaétory as might be wifhed ; Maclaurin’s refts upon the fluxional calculus, which it would be defirable not to introduce, if it could be obtained from more evident and obvious principles. Another demonftration of this theorem, and which is confidered the moft fatisfaGtory of any tbat has yet appeared, is given by ]’Huilier of Ge- neva, in his work entitled * Principiorum calculi diff. et integr. expofitio elementaris,’’ in 4to. publifhed in 1795. TINCROACHMENT, in Law. See Accrocnine. INCRUSTATION, the lining or ceating of a wall, either with gloffy flones, ruftics, marble, pottery, or ftuc- co-work ; and that either equably, or in panels and com: partiments. Incrustation, in Natural Hiflory, is one of the modes in which organic remains are preferved in the earth: thé petrifying or-incrufting matter, forming fo thin a coat of eruft around the extraneous foffil, that its external form ig nearly preferved on the ftony incruftation. Mr. Williany Martin claffes thefe among the artificial genera’ of reliquiay and obferves, ‘ Outlines,” p. 177, note, that “the common calcareous incru(tations of our rivers, &c. have beer! ranked as petrifactions ; but, with more propriety, may be con- fidered as incipient matrices."’ If the coat of mineral matter be too thick to exhibit externally the exa@ figure of the foffil which it covers, fuch are called by this author flib- inerulling matrices, of which ironftone nodules inclofing leaves of plants, &c. are well-known examples. ‘The petri- fying {fprings of Derbythire, (for a litt of which, fee Mr. Farey’s Report on that county, vol. i.) and other calcareous’ diftricts, are well known for the curious and whimiical in. cruftations which are there ina fhort time produced. INCRUSTED, or Inckusraren Colima. See Sneru/~ tated COLUMN. — INCUBATION, in Comparative Anatomy, is the term applied to the conduct of moitbirds in relting or fitting upor their eggs, in order to communicate to them the necet{iar y degree of warmth for the excitnent of their internal parts, and the developement of the fcetts, The phenomena produced by \ncubation are fo interef. 3 n “ ings INCUBATION. ing, on account of their exhibiting in a palpable manner the progrefs of foetal exiftence, that they have been ftudied by almoft every refpeCtable phyfiologiit. Writings have been left upon this fubje& by Ariftotle, Hippocrates, Fa- bricius ab Agquapendente, Aldrovandus, Coiter, Veiling, Harvey, Langly, Schrader, Theod. Aldes, Stenon, Need- ham, Malpight, Maitre Jan, Haller, Hunter, Monroe, &c. befides fome later obfervations by Blumenbach, Leveille, and other continental anatomills: Notwithftanding the labours of fo many eminent men, fome of the mott important points in the hiftory of incubation remain {till undetermined. The belt treatifes are thofe of Malpighi, Maitre Jan, and Haller, but even in thefe authors there are many errors to correét, and, except fome detached figures that have been publifhed to illuftrate particular parts of this fubject, we do not poffefs any tolerably good plates. The deficiency and doubts that ftill remain, with refpeét to feveral parts of the hiltory of incubation, prove how difficult it is to inveltigate the fubject. The membranes of the egg, and the organs of the chick, are in general fo tender, that they do not admit of being eafily or fatisfactorily examined ; and when prepared by immerfion in fpirits, acids, or other a(ftringent liquors, their natural appearances and conneions are much ob- feured. The progrefs in the growth of the membranes is found to be in many circumftances unlike what would be expected, and what it appears to be ; and the attachments of the mem- branes to each other are extremely intricate and perplexing. This part of the anatomy of the incubated egg, therefore, principally requires improvement. In the following article we have endeavoured to fupply, by our own obfervations, what was deficient in the hiftory of incubation, but, we muft confefs, not with as much fuccefs as we could wifh; there are feveral anatomical points upon which we are obliged to {peak doubtfully, and we have not been able to fatisfy our- felves refpeéting the ufes attributed to fome of the parts of the incubated egg, although we believe others will be found to be elucidated by the experiments we have made, and which are here related. Mott writers upon incubation, appear to us to have paid too much regard to the dates or periods at which they have obferved the different events of incubation to occur. Much exa@inefs with refpeét to dates is not of material confequence, and if it were} it is not praéticable, for the progrefs of in- cubation depends fo much upon climate, feafon, and the diligence of the parent, with many accidental circumftances, that hardly any two eggs agire precifely in their ftate of growth, at any given period of incubation: we are not, there- Sore, to wonder at the difference amongft writers upon this fubje& with refpe& to chronology. The proper heat for the hatching of hens’ eggsis 104° of Fahrenheit; to which degree the furface of the body of the hen will raife the thermometer wher fhe fits upon her eggs. In thofe birds that do not fit conftantly, but truft to the heat of the fun, the temperature of the eggs is probably below 104 degrees. The full period of incubation by the hen, in this country, is well known to be 21 days. Zn warmer climates it is faid to be a day or two lefs. The periods of incubation vary much in different fpecies of birds: we introduce the follow- ing table, which has been conpiled from different authors by count Morozzo, “in a letter from him to Lacépéde, to thew the periods of incubation compared with thole of the Iife of certain birds, cms Periods of their} Duration of . Names of Birds. Incubation. their Life. Authority. ' | Days, Years. Swan : 42 about 200 |Aldrovandi. Parrot - 40 100 |Wolmaer. Goofe - 30 80 or more | Willoughby. \Bagle - 30 a \Buflard - 30 € [period of life Duck - 30 not known. Turkey - 30 % eer 4 § Ariftotle and Peacock 26 to 27} 25 to 28 V Pliny. Phe ; a sl A Treatife on heafant 20 25 | 18 20 Phielews Crow ’ 20 100 or more) Heficd. Nightingale - | 19 to 20 | 17 to 18 |Buffon. Hen - | 18 — 19 | 16 — 18 |Buffon. | Din % ake mi Several obfer- ‘saa 17 18 16 17 PAIS 2 | Linnet = 14 13 — 14 | Willoughby. B A Treatife on Cz ar é net 4 eel eae uae * thefe Birds. Goldfinch - 13 — 14 | 18 — 20 |Buffon. Crane and heron, as well as offrich, hatch their eggs chiefly by the heat of the fun. Journ. de Phyfique Ventos. an. ro. The ufual mode of examining the progrefs of incubation, is to gently fracture the fhell, and pick it carefully off in pieces with a pair of forceps. This fhould be done at the large end, as it is there the changes of incubation commence, and at all periods it will be found the meft convenient place for breaking the fhell, on account of the air-cell being inter- pofed between it and the fluids of the egg. For particular purpofes other modes of preparation mult be employed. If it be our obje& to ftudy the formation of the membranes, thefe maft be feparated from the humours, and after being cleanfed by being paffed through feveral waters, they are to be floated in weak fpirits over a blue paper, or fome other dark coloured fubftance. The flrongett fpirits, diitilled vinegar, or a folutios of alum, are neceffary to render the organs of the foetus fufficiently plain for examination. The only ufe of boiling the contents of the egg, is to fhew the relative fituation and magnitude of different parts. In many obfervations, it is neceflary to magnify by glafles the fubjeéts of inveftigation; we would however recommend, that the fight be only affifted by the fingle lens. Whena higher power is employed it is liable to deceive. After the egg has been fat upon for 12 hours, the quantity of air contained between the two layers of the membrane that lines the fhell, is found to be a little increafed. No evident change is yet to be obferved with refpe& to the bulk or confiltence of the Aumours. The white {pot on the fuperior furface of the yolk, which contains the cicafricula of the ancient anatomifts, or the follicle of Haller, is now fomewhat enlarged, There appears, in its interior, a fmall white body of an elongated and in- diftin& figure, which is furrounded by a line of a deep yel- low or greenifh colour. If this part of the membrane be removed and examined with a lens on the under furface, after being a fhort time immerfed in {pirits or vinegar, it will be difcovered that the elonvated white body is the rudiment of the fetus, which is already inclofed in the amnics. The greenifh appearance would feem to depend partly upon the exiftence of the amnios, aud partly upon a depreflion on the fubftance -INCUBATION. fobftance of the yolk, which is the origin of the bed of the chick. We cannot determine whether this depreffion con- tains a fluid or not. The part of the white {pot which fur- rounds the depreffion that contains the fcetus, is occafioned by the fubftance of the yolk having a peculiar organiza- tion. At that point it is white, and has a curdy or coagulated appearance. Malpighi believed that he could difcern the foetus as early as the fixth hour, and even before the egg had been fat upon; but molt other obfervers have not been able to fee any appearance, which they thought ‘deferved to be confidered as the feetus, before the 12th hour. It is, however, probable, that the embryo of the chick is formed after impregnation, although from its extreme ten- dernefs and obfcurity of figure, it cannot be deteted until it is a little developed by incubation. The amnios appears to be coeval with the chick. The rings or circles (halones) that furround the cicatri- cula, are now fomewhat enlarged, and more delined in their outline. ‘They alfo affume a little of an oval figure. ‘lhe external circle is undulating, or forms waves, through which the colour of the yolk appears. Maitre Jan defcribes four halones, Haller only three; we have found the number vary. This appearance of circles is not eauly accounted for. We have thought it arole from the feparation of the membranes into their proper layers. Firft. Day, or after 24. Hours—Upon opening an egg at this period, it is perceived that the fpace formed for the admiflion of air between the layers of the external membrane is a little more enlarged. The external white appears fcarcely diminifhed ; it is fluid and tranfparent, and runs out at the opening if the egg be inclined to that fide. The yolé is found ‘to float higher in the principal white than before incubation, and appears a little flattened upon the upper furface. The halones are more extended, particularly the outer one. The internal one has acquired a little of a heart-fhape, and is furrounded by a more defined line of a greenifh colour, which is the ficlt appearance of the outline of the veinous area. There are alfo fome faint and imperfeét traces of green veflels, or rather elongated points, to be deteted in different parts of the {pace inclofed by this line, which are the rudiments of the veilels that are to compofe the area. _ The @mnios is well formed, and compofed of two circles which unite two ftraight lines, refembling the figure of a eftle. 7 The fetus has a more determined figure ; it is larger at both extremities than in the middle ; but the end which cor- refponds to its head is larger than the other, and forms an eval mafs. Maitre Jan ftates that if an egg, after being under the hen for 24. hours, be put into water, it will float with the {mall end downwards, which is a proof that a part of its fluids is loft, and replaced by the admiffion of air into the air-cell at the great end of the egg. _ After 36 hours we have found the greenifh line, already mentioned, to have {till more plainly the appearance of a veflel. It encompaffes a larger area, and aflumes more com- pletely the contour of a heart. The greenifh points alfo now are continued into each, and put on the appearance of veffels, which are moft numerous next the circumference of the area, but prefent no evident conneétion with the fetus. Some of thefe veflels are feen to contain red blood, more par- ticularly thofe which pafs between the head of the chick and the upper part of the veffel in fhape of a heart. In fome eggs, at this period, we have found thefe red veffels approach the chick but not joia it: in others they were collected by a fing'e branch, which pafled behind the head of the chick. This branch Leveille has called the meningo-cardiae veit, and others the vena afcendens. There is confiderable va- ” riety with refpect to the period at which red veffels begin to appear. We have feen them before the 36th hour, but oftener, perhaps, not before the goth, or even later. Authors differ extremely or this point. Malpighi reprefents the commencement of the veffels at the goth hour (Append. figs. r8 and 19.), and even delineates them at the 12th hour. (Epift. r. fig. 5.) Stenon obferved them at the 71{t hour ; Vefling at the 43d, but faw the pcints at the 42d hour. Aldrovandus found them the third day. Maitre Jan de- {cribed the firft appearance of veflels at the 38th hour, and reprefented them diftin@tly formed, and carrying red blood at the 44th hour. (P. 53 and 58, and fig. 10.) Haller noticed fome reddifh points at the 36th hour, and to the 48th, but did not perceive perfe&t red veffels before the end of the third day. Memoire ad fur la Formation du Coeur dans le Poulet, &c. p. 23. Thefe contradi€tory accounts depend in fome degree upor their authors net having fufficiently attended to the exiftence of colourlefs points and veflels before the formation of red blood. Haller ftates, that he obferved, at 313 hours, the head of the fetus to have the appearance of being fplit, which he attributed to the veficles of the brain having a {mall degree of opacity, whilft the membranes compofing the cranium were fo tranfparent as to be feen through. Malpighi alfo reprefented this appearance. We have not noticed it. Haller likewife defcribes the head of the chick to begin to form at a right angle with the tail at the goth hour. After 48 hours, or two days, the external halones occupy a confiderable extent upon the furface of the yolk, and be- come fainter in their appearance. The liquid or external white is fomewhat diminifhed. The principal white begins to fublide to the bottom and {mall end of the egg. The yolk-bag appears flatter upon the upper furface, and its contents are a little more fluid than before incubation. The chalaze, which were originally fituated at the two poles of the yolk-bag, are now found {omewhat below them, and confequently rather nearer to each other. The vafcular area (figure veineufe of Haller) is now in ge- neral completely formed with colourlefs and red veffels. The number of the latter is very variable. We have found them at this period in very different {lates of advancement, but always difcoverable by the greenifh veffels which precede them. ‘The area has very perfectly the figure of a heart, except that the part correfpondiug to the bafis is formed by the outline of the two fides being confiderably curved before their junction in the middle, at which place the outline of the area forms a pointed projection downwards, which nearly reaches to the head of the foetus. It is this part of the area which receives the branches of the meningo-cardiac vein already mentioned. ‘The veffels are now alfo connected with the chick by means of their principal trunks. "The pulfation of the heart is evident at this period, and even in fome hours before, according to Maitre Jan’s ob- fervations. It appears to be produced by the alternate contraction and dilatation of two or fometimes three veticles. This appearance is the pun@um faliens of the older anatomitts, and has been commonly, but erroneouf}y, reputed the firlt action of life. One mode of preparing eggs for examination is to remove a portion of the membranes including the feetus and vafcular figure around it, and having wathed off any pieces of the yolk that may adhere to them, immerfe them for a fhort time in ftrong fpirits or diluilled vinegar, by which means the pulpy INCUBATION. pulpy fubflance that compofes the chick is rendered opaque and vilible, and the itruéture of the membranes more dif- tinguifhable. If this flep be taken with the egg at the prefent period, or even before it, a fine glofly tranfparent membrane will be found upon the external furface of the yolk-bag, which is very readily detached, leaving the foetus and the vafcular area beneath. This membrane, we conceive, is the fame which, ilrengthened by the velicula umbilicalis, becomes the one which at lait entirely envelopes the ¢ontents of the incu- bated ere. The membrane, containing the va/cular area at the fecond day, may be diftinguifhed to be compofed of two layers ; the external is {mooth, and receives the veflels ; the internal is alittle thickened and fpengy. ‘fhe remaining portion of the yolk-bag is not feparable into different layers. From hence it would appear that the divifion into layers is the confequence of vellels being diltributed to the membrane. The amnios is now found a little larger, and the tranf- parent fluid with which it is filled in greater quantity. The fuperior circle alfo, or that which inclofes the head of the chick, is larger than the lower. The fetus has altered its fhape a little. The head has acquired fomething the figure of the ace of clubs, the an- terior prominence of which correfponds to the nofe and mouth ; and the two other projections are the veficles con- taining the brain in-a very foft pellucid ftate. Maitre Jan ftates that the eyes alfo are vifible at this period. (P. 77.) Haller did not fee the eyes before the sift hour, at which time they had not any colour, and were merely diftinguith- able from the other parts of the head, by their prominences and fituation. Sed. xii. Second Memoir. The eart, before the chick is removed from the membranes, and rendered opaque by immerfion in fpirits, appears to be fituated on the outfide of the body, and the pulfating points feem unconneéted with each other ; but upon aclofe exami- mation it is feen that the heart is not uncovered, although, from its projection, and the natural tranfparency of the parietes of the cheit, it had every appearance of being fo. The heart altogether refembles a horfe-fhoe in form, or a portion of a circle, of which the convexity is turned for- wards, and the limbs backwards, one being direéted to the upper part of the cheft, and the other to the lower. The heart at this time confiits of an auricular canal, a ventricle, and a bulbous enlargement belonging to the aorta. ‘The two Jait are the falient points, and when the chick is fo far ad- vanced that the auricle is formed and pulfates, which is fome- times the cafe at the 48th hour, there are three falient points to be feen. Maitre Jan defcribes four falient points, and fuppofes that they are the rudiments of the two ventricles and two auricles of the heart. At other times, he fays he has fometimes feen but three pulfating points, or perhaps two, er only one. Haller, who paid particular attention to the developement of the heart of the chick, contradiés Maitre Jan’s account. At the 48th hour he could difcover but two falient points ; at the 5oth hour he faw three, but on no occafion more than that number. He fuppofes that when there are two falient points the auricle does not pulfate, and when only one.the bulb of the aorta is without motion. Me‘noire 1. p. 83. The End of the third Day, or Seventy-two Hours.—This is a very important period in the hiftory of incubation. The feparation of the layers of the external membranes for the admiffion of air continues to increafe. The halones are fearcely to be difcovered. The yolk is more extended, and the white continues to gravitate to the lower parts of the egg. The va/cular area-is now generally found to be perfected with red veffels. The contour of the area is more extenfive, particularly in the tranverfe direction, The diameter being now about an inch and a quarter, it has therefore lefs the fhape of a heart: the pulfation of the three veficles, and diftribution of the blood-veffels, are very clearly feen through the tranfparent membrane which covers them. The amnios begins to affume the figure that it is after- wards to poffefs. The two circles which encompafs the head and the tail of the foetus become enlarged, principally on the fore-part, or the fide next the right-hand of the ob- ferver ; but the fac is a little contracted on the fame fide oppolite to the umbilicus of the chick, thus tending, Ina degree, to give the amnios the form of a kidney. The fetus has a curved pofition forwards, and its body appears larger in proportion to its head, than on the pre- ceding day. : We have forborne to defcribe particularly the Aeart and blood-veffels of the chick and area until the prefent period, when they are more diftinétly feen. In order to difcern perfectly thefe parts, the membrane including the foetus and vafcular area fhould be removed from the yolk-bag, floated in clear water, and examined with a lens. The vafcular area contains four or five principal veinous branches, which ramify-upon the membrane, and produce a very clofe and intricate net-work towards the cir- cumference of the area. Thefe branches appear, when the chick is in the natural fituation, to be fo many diflin& trunks; Two of them are feen to come from the fides of the chiek, a third from under the tail, and one, or fometimes two, branches from under the head of the chick. The two firft are commonly called the umbilical veffels, as they come out of the chick at the umbilicus: they would, perhaps, with more propriety, be called mefenteric veins, as we have much reafon for fuppofing that they are the fame which are fo plainly branches of the mefenteric vein in the latter flares of incubation. At the prefent period they terminate in the vein which correfponds to the vena cava and vena porte. Both the branches which emerge from undes the tail and head of the foetus are found to be given off from the umbi- lical veins juft outfide the body of the chick. ‘his origin is. concealed from view until the lower furface of the membranes and chick are carefully examined. Several anatomifts feem to have publifhed errors with refpeét to the fuperior or afcending veffels of the vafcular area. They are often de- f{cribed as diftin&t branches, which are loft on each fide of. the notch or chink, which has been already mentioned to, exift during the firft days of incubation in the contour of the vafcular area. They were, therefore, fometimes fuppofed to be feparate veffels. Langly reprefented them to be at firft diitinét branches, which united at the 61ft hour; but Malpighi, Maitre Jan, Haller, &c. ftate that thefe two branches always unite into one behind the head of the’ foetus when they do not fooner. Malpighi, Maitre Jan, and others, however, reprefent this veffel as coming from the heart: Maitre Jan even fuppofes it may be the afcending aorta. Our obfervations, neverthelefs, lead us to agree with Haller refpecting this veffel, who ftates it to be a branch of the umbilical vein. It is this veffel to which Leveille has given, as we conceive improperly, the name of meningo-cardiae vein. The veflel which forms the contour of the vafcular area - has beencommonly defcribedas a fingle vein: Malpighi, how. ever, reprefented it as being compofed of an intertexture, or plexus of the ultimate branches of the veins which are fpread over the vafcular figure. ‘There appears to be fome founda. tion INCUBATION. » tion for both thefe accounts, according to our obfervations : thus we have fometimes feen with a lens the appearance de- {cribed by Malpighi, and at the fame time the fingle trunk nearly tranufparent from wanting red’ blood, and enveloped and conceated by the intricate and clofe plexus formed by the extremities of the veins of the external part of the area. We believe, however, that there is in every cafe originally a fingle veffel which makes the outline of the veinous figure or area, and which exifts previoufly to the veffels which com- municate with the chick. Jt is not uncommon to find this veffel deficient at fome parts of the circumference of the area, more efpecial y at the fuperior part, where the outline makes the pointed projeCtion inwards towards the head of the chick. Sometimes a vein is feen to pafs acrofs the chink formed by the outline at this part. The arteries which ¢o to the vafcular area bear no propor- tion to the veins. They are fo fmall that they are not ealily feen, and even have been by fome {uppofed not to exilt. The left umbilical artery is in general much more contider- able than the right; indeed we have fometimes doubted whether the latter exifted or not. They accompany the veins in their courfe, but are not fuppofed to have any imme- diate connection with the veffel which forms the contour of the area. Betlides the fimail fize of the arteries at this period of incubation, they are difficult to difcover on account of all the blood-veflels of the egg carrying blood of the fanie co- Jour. During the firft days the blood is a bright feariet, or poffeffes an arterial character. The parts compofing the heart of the chick, although formed feveral hours before, are now particularly evident. The vein, which in the firlt periods conveyed the blood from the vafcular area to the _horfe-fhoe dilatation correfponding to the heart, is now feen to be the vena cava, or rather it fhould be confidered as the vena porte and cava together. At the fuperior part of this vein a dilatation is formed, which is prolonged tranfverfely forwards, and inclines alittle downwards. This dilated part, in which the vena cava ends, is the origin of the fingle auri- cle of the heart of the feetus, and is the firft veficle in order which is feen to pulfate. Towards the end of the third day, according to Haller, the enlargement of the head of the vena cava becomes greater, and is diftinguithed from the reft of the vein by a whitifh- coloured interval; it is ftill, however, but a fingle cavity : its greateft diameter is acrofs from the right to the left hde of the foctus. The canal which conveys the blood from the auricle into the veatricle (/e canal auriculaire ot Waller), exilts, as already mentioned, before the vena cava becomes dilated to form the auricle, at which time it might be con- fidered as the continuation of the cava. This canal is, there- fore, one of the firit formed parts of the heart. It is moit Jong and flender when it firit appears, and gradually con- tracts in length, until it is loit in the fubitance of the heart. lt is fmalle! where it arifes from the auricle, and widelt where it opens into the ventricle ; its diameter, however, is always lefs than that of the latter. It enters the ventricle on the left fide, for which purpote it defcends a little, and therefore croiles the aorta, which arifes from the ventricle on the right fide and afcends backwards. There arifes from the decuffation of thefe two veflels the appearance of a kind of knot, and the auricular canal feems to double upon itfelf, in order to be continued with the aorta in the oppofite and concealed fide of the knot, When the auricular canal is viewed on the right fide, its extremity is concealed, and the manner in which it opens into the ventricle cm perceived: it-feems rather as if {eparated from the ventricle by a line which is the border of r ~ the heart. But if it be viewed on the left fide, there is ne difficulty in difcovering its communication with the ven- tricle. The ventricle is a fingle cavity during the firft four or five days of incubation; or rather, the left ventricle exifts only at that period. When the two ventricles make their appear- ance, it is in confequence of the right ventricle being added to that which had already been formed. The left, or fingle ventricle, arifes out of the ring-fhaped veffel, which con- ftitutes the firit appearance of the heart, and probably is the firit point which pulfates. It is an oval veficle, with tranf- parent parietes in the commencement ; but it foon acquires more of a conic figure, with a pointed apex, and greater thicknefs and mufcular power : it is, however, very nearly oval at the prefent period. The aorta, as already mentioned, is coeval with the firit appearance of the heart, and exits before the formation of the auricle. It is then very long, and its arch is much greater in proportion to the length than in the fucceeding periods of incubation, as in the adult bird. It makes an extenfive curve backwards, in order to enter the cheft, im- mediately under the head, and compofes a confiderable part of the ring which reprefents the heart. On the left fide this artery appears as if cut off by a line which terminates the heart. The auricle, when formed, is {urrounded by the arch of the aorta, which is fuperior, and more for- wards. A few hours after, the part of the aorta which appears to be outfide the cheft, becomes fhorter, and fome inequa- lity is perceived in its diameter. ‘The part of the aorta, which immediately arifes from the ventricle, is more con- tracted than any other; and the wave of blood which is feen to pafs through it is very flender. Haller gives the name of the frait (le detroit) to this part. The next part of the aorta is a dilatation, to which Haller has given the name of the du/d of the aorta. This portion of the veffel, as already mentioned, conttitutes one of the pulfating points, and is feen to beat even before the auricle. The bulb of the aorta has been miltaken by Malpighi for the left ventricle, which, confidering its lively and early pul- fation, was not unnatural. He likewife was led by the fame error to take the fingle ventricle, which is a€tually the left for the right, or pulmonary ventricle of the heart. The third portion of the aorta is cylindrical; its direc- tion is changed, and it reprefents with the preceding a point or beak. It is fmaller than the bulb, but larger than the branches which go to produce the dorfal aorta. Haller calls this part the bec de Paorte. Soon after the prefent period, and before the end of the fourth day, there are branches feen to’ arife from the beak of the aorta, in order to form the dorfal portion of that veffel. “ae The three parts which compofe the heart of the fetal chick pulfate fo rapidly, when examined immediately after the egg is opened, that the eye cannot follow them; but in proportion as the chick becomes cold and feeble, the inter- vals between the pulfations become longer. The pulfations: may be thus gradually reduced to fifteen, to ten, to five, or two, or at Jalt even to but one in a minute ; neverthelefs, the celerity with which each contraCtion is performed is {carcely diminifhed. When. the pulfations of the heart are rendered fufficiently Qow to be counted, they are perceived to follow each other in a regular fucceflion. The auricle always beats the firtt of the three veficles. In its {yftole the point defceids, and the fides are approximated ; it {carcely ever appears to expel 5 entirely INCUBATION. entirely the drop of blood it contains, and therefore it re- tains almoft conitantly a degree of red colour. The wave of blood which proceeds from the auricle ap- pears to vanith, as one cannot fee its courfe into the ventricle unlefs the heart be beheld obliquely from above ; then the blood will be feen to pafs along through the auricular canal, which, as before obferved, is concealed by the ventricle. ‘The contra¢tion of the ventricle fucceeds to that of the auricle: it approximates its fides, and becomes narrower and fhorter. ‘The ventricle frequently becomes quite pale during its contraction, owing to the entire expulfion of its contained blood. After the contraction of the ventricle follows that of the bulb of the aorta, which pulfates as quickly and perfeétly as the ventricle itfelf. It empties itfelf completely at each contraction, and therefore appears quite pale at thefe moments. The pulfatory movement terminates at the origin of the three veffels which go to form the dorfal aorta, and this vef- fel declines very much in bulk oppofite to the umbilicus ; the branches which go to fupply the inferior parts of the feetus being very infignificant. The different membranes become more diftinG at the end of the third day than they were before. At this period alfo the firt appearance is feen of that remarkable membrane, avhich is called by fome allantois, and by others veficula um- bilicalis. When a portion of the membrane is removed containing the vafcular area and the chick, and wafhed and fpread out in clear water, it is found to contain four layers. The firit is thin, tranfparent, and fhining, and refembles that which forms the yolk-bag before incubation, It very readily fe- parates from the other layers. The fecond is alfo tranfpa- rent, and ftill thinner than the firft, but is more adherent to thofe underneath it. The third is the membrane in which the blood-veffels are diftributed that form the vafcular area ; and the fourth is the internal or fpongy membrane of the yolk-bag.. The third membrane feems to grow with the wafcular area, by which it appears to be bounded : the inter- nal membrane of the yolk-bag is at this period a little {pongy or villous at every part, but much more fo under the wafcular area than at any other place, The chick inclofed in the amnios is fituated between the two firft and two lait of thefe membranes. The vefcula umbilicalis,; when firlt difcovered, makes its appearance as‘a {ma!l fpherical tranfparent bag, which pro- jects out of the opening into the abdominal cavity of the ehick, that is left by the reflection of the. amnios inwards. At this period, and for fome time afterwards, it is diflended by a clear fluid. The end of the fourth day, or ninety-fix hours, the ap- pearance of /alnes is in general no longer to be perceived. The vajcular area is fo much extended that it occupies near the half of the fuperficies of the yolk: the outline has no remain$ of the heart fhape, but has aflumed an oval figure, the greateft diameter of which is in the tranfverfe direction of the chick. The air-celf continues to enlarge at the great end of the Lge. ‘The external or liquid white appears to be all confumed. The principal white is chiefly to be found in the lower parts of the egg, and when poured out is feen to adhere a jittle to the internal layer of the membrane of the fhell at the {mall end of the egg. The yolk-bag is {ll flatter in its figure than before, and ts contents are evidently more fluid than they were previous “fo incubation. The chalaz@ continue to approach each other below thé poles of the yolk-bag. The amnios is confiderably increafed, contains more fluid, and forms a deeper bed for the chick on the membrane of the yolk-bag. The weficula umbilicalis has made a rapid progrefs fince the end of the third day. It now projeéts confiderably beyond: the feetus. It appears flatter, and lefs full of fluid than at firft ; and at this period, by examination with a magnifying glafs, a veflel may be feen to ramify upon it. Upon opening the yolk-bag, and wathing off the yolk, the fpongy coat is feen to be elevated in the courfe of the blood- veflels of the vafcular area: this is the firft appearance of the valvular itruéture of the yolk-bag, of which more here- after. The fetus is till more curved in its form, the prominence corre{ponding to the bill being bent down upon the breatt, The rudiments of the upper extremities are apparent; they are two white buds or fhort projections. ‘The lower limbs are likewife produced, but without any more form. The cerebral velicles are more eminent 3 and, when opened, their contents are found, though {till pellucid, to be mixed with fome whitith fibres, which form opaque clouds, and are the commencement of the brain. Haller found the Acart beginning to undergo fome changes about this period. he auricle, when viewed in the ufual manner on the right fide of the foetus, is feen to contain two femicircles. The mott pofterior of thefe, or the one neareft the vertebra, is the largeft, and terminates the auricle. The anterior femicircle is fmaller, and divides the auricle as a kind of feptum. This is the firft appearance of the diitin@ion of the auricles into two. That. which is to be called the left is the larger, and projects over the right potteriorly, The latter, which is the fmaller, has a femilunar termination, which is comprifed in the breadih of the left auricle. The vena cava and the auricular canal are inferted in the interval of the two auricles. The diftin&tion of two auricles is likewife to be feen on viewing the heart on the left fide. That which is to become the left is oval, wider than the other, and pafles beyond the auricular canal. At the end of the fourth day, or foon afterwards, the two auricles become more feparate from the vena cava and the auricular canal. They projeét over thefe veffels, and are elevated from the two fides of the bafe of the heart into a pointed top. The whole of the auricle, or the united refer- voir of the right and left auricles, appears then as a fac with two pointed horns: each of thefe pointed tops has a fepa-° rate drop of blood. 'The border of both the auricles ap-' pears at this time notched. Mem. 2d fur fa Formation du Coeur dans le Poulet, &c. par M. de Haller, p. 70. The auricular canal, towards the end of the fourth day, pafles more in a dire€tion from above downwards than for- merly. “ According to Maitre Jan, the rudiments of the /ungs are vifible at this period ; but Haller afferts that they do not appear before the 138th hour. ‘The former author defcribes at this time, in the neighbourhood of the heart, little reddith clouds, apparently of a {pongy nature, which appear more or’ lefs red, alternately, as the heart moves, and which remain of aa uniform colour when the heart has ceafed to beat. Maitre Jan admits that the pulmonary veffels cannot yet be difcern- ed, but fuppofes from the above effet of the heart’s motion, that they do exift, although they are not vifible. (Obfervat. fur la Formation du Poulet, par Maitre Jan, p. 115 and 122.)) —Not having examined this point, we cannot. decide between: thefe authorities, byt we fhould judge it improbable that the pulmonary INCUBATION. pulmonary veffels could regularly tranfmit blood to the lungs, and yet not be vilible. ; : Both Haller and Maitre Jan agree that the ver makes its appearance towards the end of the fourth day. It is feen as a vifcous cloud of a reddifh yellow. colour, fituated between the umbilical veffels and the defcending aorta. It is render- ed more vifible and opaque by moiltening it with diftilled vinegar. vith ty tate 3 . The. eye of the f. appears black, the choroid coat ‘being formed; and theltipesor part of the eye chiefly pre- fenting itfelf to view, the pupil is fituated in the inferior portion of therye. - bed The end of the fifth day, or 120 hours, the vafeular area is fo much extended as to cover generally more than half the fuperficies of the yolk-bag. The trunks of the veffels are proportionally increafed in fize, but the branches near the circumference are not fo much advanced. Some additional veffels are feen to proceed from the umbilicus of the chick towards the left fide of the area. Thefe appertain to the ve- ficula umbilicalis, as will be better underftood hereafter. All the blood-veflels of the area, which were in the firft days of incubation of a fine {carlet, are now commonly found of a dufky red colour, ; The amnios continues to increafe. in magnitude, and con- tains fo much fluid, that the chick appears to be furrounded by a green liquor. The chick, efpecially when excited by a little external heat, can be perceived to move its limbs. When the external membrane is removed, the veficula umbilicalis is found to be confiderably larger than it was the preceding day. It is likewife perceived to have contra¢ted fome adhefion with the external membrane, which however is undone by a flight force, particularly after the membranes have been a fhort time immerfed in vinegar or fpirits. The branches of blood-veflels which are diftributed upon the velicula, belong to a trunk that comes out through the opening of the abdomen at the umbilicus of the chick. The veficula is connected with the foetus by means of a neck, or contracted part, which is likewife received into the abdo- men at the umbilicus. Maitre Jan defcribesat this period a number of fine branches of blood-veffels upon the fkin of the chick: we have not, however, obferved thefe fo early. When the internal furface of the yo/é.bag is examined, it is found that the eminences upon it, already mentioned, have affumed the form of veffels ramifying from the centre, where the chick is fituated, to the circumference of the vafcular area. ‘This appearance has been miltaken by many of the older anatomilts for real veflels diftributed upon the inner furface of the yolk-hag, and were fuppofed by fome to be defigned to convey the yolk to the chick. When the appear- ance of the yellow veflels of the yolk-bag is firft prefented, it is very diflicult to detect the real occalion of it, but as it advances, it becomes plain that the effect is produced by the inflection or doubling of the inner membrane of the yolk- bag, iaithe courfe of the real blood-veflels of the bag. By a careful examination of the matter witha lens, even at this pe- riod, we have always perceived the appearance of yellow vellels to arife from the blood-veffels running in grooves, or reflections of the inner membrane. When the two tunics of the bag are feparated, the blood-veffels are found to belong to the exter- nal, upon the inner fide of which they appear as eminent lines ; while upon the external fide of the inner coat there are correfponding grooves or impreffions. The deart at this period begins to undergo the changes by which it approximates in itructure the bios of the adult bird. It. becomes perpendicular, or in the line of the foctus, Vou. XIX. from being placed in a tranfverfe direCtion. curved between the two lobes of the liver. The auricks become more diltin@, and the interval be- tween them white. The auricular canal is confiderably fhortened. The fecond or right ventricle begins to be formed before the termination of the fifth day. It appears as a little fac under the bulb of the aorta. It is much fhorter than the original ventricle, above which it is placed. Its figure is oval, and ithasa red colour. Ina fhort time the new ventricle is feen in its proper fituation, that is, on the right fide of the heart. Haller ftates that he found it placed on the right as early as the 117th hour, There are then two diftin& drops of blood in this part of the heart of the faetus, and a white line is per- ceived between them. The aorta aries from the left ventricle. The branches, which are furnifhed in order to form the dorfal aorta, have an inclination downwards, and unite in an acute angle. At this period alfo the bulb ef the aorta begins to diminifh, and to.retire into the flefh of the heart. The iver is more diftinét than at the end of the preceding day ; it is divided into tobes, and is of a pale or yellowilh red colour. Its vefiels receive blood. Fialler firft perceived the re€tum at this period, although the {Ltomach and other inteftines were not yet formed. It ap- pears in the fhape of a trident, the lateral procefles of which are produced by the rudiments of the two cceca. The End of the 6th Day, or 144 Hours.—Eggs placed fo long under the hen are confiderably diminifhed in weight, which is even perceivable on taking them into the hand. ‘The fhell begins to appear more dry, and opaque coloured, and is more fragile than before incubation. The diminution of weight depends upon the enlargement of the air-cell, which now extends over a confiderable part of the great end of the egg. Maitre Jan has taken fome pains to prove that the air contained in the egg is not derived from the atmofphere. He fuppofed that the fluids filtered through the inner layer of the membrane which lines the fhell, and that by the aétion of heat, they were volatilized or converted into air. This opinion is not only improbable, but refuted by particular fa&ts. On opening the air-cell, we do not find its furface to poffefs any moifture ; that layer of the membrane which lies next the humours of the egg, al- though lefs thick, is fmoother, denfer, and lefs porous than the layer immediately in contact with the fhell, and therefore lefs fitted to admit the tranfpiration of either fluids or air. The chemical properties of the gafeous‘contents of the air-ce!l feem to put it beyond all doubt tlt the atmofpheric air pailes into the egg. (See the article Ecc.) We fhall only add, that when we have varnifhed the externa! furface of the fhell the air-cell no longer increafed in fize. Upon opening an egg at this period of incubation, it is not unufual to fee the chick turned upon its back, inftead of lying, as before, upon its left fide. ‘This circumftance feems to depend upon the amnios having fo much increafed in fize that the foetus may roll round within it. The membranes around the chick begin to acquire more ftrength and firmnefs, which is the eancence of the vefi- cula umbilicalis becoming more clofely and extenfively united to the external membrane. The weficula is now found to be very much inereafed in mag, nitude, and to cover a confiderable portion of the yolk-bag. When the membranes furrounding it are cut open, and the velicle expofed, its neck or peduncle may be traced as a fine tran{parent tube to the rectum, by a careful examination, particularly if the parts have been, rendered opaque, by > beg The point is re= INCUBATION. being a fhort time fteeped in vinegar or ftrong fpirits. Where it opens into the retum it is dilated. The connection that exifts between the veficula umbilicalis, and the end of the rectum, by means of a tube analogous to the urachus, has induced Haller and others to confider this bag as the refervoir of urine. We thall find in the fubigquent periods of incubation, that the veficula umbilicalis performs very important and other functions than thofe of a refervoir of urine, in which, never- thelefs, it does not perhaps materially differ from the allantois of quadrupeds. It is neceffary here to notice an error which two of the belt writers upon the fubje& of incubation have fallen into with re{pe& to the veficula umbilicalis. Both Malpighi and Maitre Jan fuppofed it to be a part which afterwards be- came the ftomach of the chick. ‘This miftake appears to have been occafioned by the veficula becoming, as we fhall afterwards deferibe, fo much extended as to lofe the figure of a bag, and from being united to other membranes, only to be diftinguifhed with difficulty. Its being vifible nearly three days before the ftomach ; acquiring even then a much greater fize than that vifcus pofleffes, and differing effentially in figure and texture, fhould however have prevented thefe two parts from being confounded with each other. The chick, on the fixth day, has afflumed a good deal of its natural form. ‘The extremities are fo much developed, that the divifions of thefe members are vifible. The wings have their bent figure, andthe legs are added to the thighs. This period is likewife marked by the firlt appearance of moft of the important organs of the body, befides fome changes in thofe already formed. Haller ftates that he could difcern the pericardium at this time. The two auricles are fill more diftin& than at the end of the preceding day. The auricular canal, which had before begun to retire within the flefh of the heart, is now not vilible. The ventricles are better formed, and broad at their bafe. The bud, the frait, and the Jeak of the aorta, which gave origin, in the early periods, to the principal arterial trunk, are in general not apparent at the end of the fixth day. This veffel is feen to come forth from the bafis of the heart ifelf. The pulmonary artery was feen by Haller firft, about the middle of the fixth day. This veffel is lefs firm and opaque than the aorta; it arifes from the left fide of the bafis of the heart with a little bulb. It is ufually of lefs diameter than the aorta. Almott at its exit from the heart it divides into two branches. The branch on the left fide forms an arch, in order to reach the lung of that fide in paffing behind the bronchia : it joins the aorta, towards the third vertebra of the back. The branch of the pulmonary artery, which is inferted into the dorfal aorta, 1s longer than the dud@us arteriofus of quadrupeds, to which it is analogous. It fhould, how- ever, receive the name of £ff, as birds have a fimilar ar- terial duct upon the right fide. ‘The left lung receives a con- fiderable branch fromthe dutus while on its way to join the aorta. The other principal branch of the pulmonary artery paffes tranfverfely to the right fide, under the aorta; to which it adheres. ~It furnifhes the artery to the right lung, and defcends behind the bronchia, in order to open into the aorta, bigher than the left branch already mentioned, The right arterial du& is much fmaller than the left. ‘Thefe arterial ducts are two of the three branches, which have been defcribed in the preceding days as contributing to form the dorfal aorta. The right dué& is larer in ap- pearing than the other, and hence, when the aorta is firit difcovered, it is obferved to be only formed by two branches. About this period the /ungs have fufficient confiltence to be viible. When they are rendered opaque by vinegars they are found to be not fo fmall, that they might not have been feen fome time before if their tranfparency had per- mitted it. Their figure is cylindric. ‘hey aré attached polteriorly to the back, and conne&ted with the two fides of the heart. The /iver has acquired more of its future form than it had the preceding day. The fomach is apparent at this time. It is very thin and tender, and refembles a good deal the form of the human ftomach ; its orifices are, however, nearer to each other. The ftomach is received between the lobes of the liver. The fluid contained in the cerebral veficles has acquired by this time fome confiftence. The kidnies are difcovered during this day as long reddifh ranulated lines or traces in the loins. The End of the 7th Day, or 168 Hours.-—-Upon opening an egg at this period, we perceive that the changes already mentioned in the form, confiltence, and fituation of the hu- mours have continued to increafe. The white tends more towards the lower part and fmall end of the egg: it continues to diminifh, and to become more tenacious in its compolition. The yolé is more fluid, more fread out, and apparently of a larger fize than before. Haller neverthelefs found it at this date to poffefs its original weight. : The blood-veffels which are diitributed upon the membranes, and which were in the firlt days, as before mentioned, of a brilliant {carlet, are now in moft eggs found to be all of the purple or venous colour. The uniformity of colour makes it difficult to diflinguifh the arteries: they can in- deed be only afcertained by diffecting the membranes and purfuing the courfe of the veffels into the body of the chick. The amnios is fill larger at this than any, previous pe- riod; it preferves its kidney form, and its fluid is tranfparent, and cannot be rendered opaque, or coagulated by vinegar or fpirits. The veficula umbilicalis goes on increafing im fize, and its’ veffels acquire f{trength. The chick has its members more developed. The feet are added to the potterior extremities, and even the foes can be difcerned. Offification has commenced. Haller ftates that he prepared the femur and tibia at the 146th hour. The tevo mandibles begin to form at the under part of the head, leaving an aperture between them, which is to be the mouth. Haller at this period obferved the firft convolutions of the /mall inteftines, which were fo extremely tender as not to admit of being touched, although the rectum was appa- rent two days betore. The /pleen now firlt makes its appearance. Haller fup- pofed that he could likewile difcern the renal capfules and OVaTTCS. The intermediate of the three veflels which arife from the heart to form the dorfal aorta, is, about this time, feen to furnifh the ¢qwo carctids; it 1s the fame veffel which. be- comes the de/cending aorta, when the ftruéture of the pul monary artery 1s completed. Haller ftates that the puifa- tion of the aorta ceates after the 154th hour. The eye of the fotus is contiderably developed at the prefent period. he three humours.exift; the witreous is 3 extremely ss INCUBATION. extremely abundant, tpon which depends the great bulk of the eyes of the chick about this time. The /ens is fmall. The iris is formed, but is very tranfparent. ‘The trunks of the optic nerves and retina are vifible. The latter is tran- {parent and plaited, and feems to go as far as the lens. The ‘radiated black lines which fornt the corona ciliaris are feen ie the retina, placed around the external part of the ens. The End of the 8th Day, or 192 Hours.—The vafeular area hhas increafed fo much by this time, that it very nearly covers all the fuperficies of the yolk-bag, except what is in contaét with the white, The branches of vefiels, how- ever, begin to decline or difappear in fome parts, particu. larly towards the circumference, although the trunks are larger than at any former period. The blood-veffels belonging to the veficula umbilicalis are confiderably enlarged and extended, and are now feen to turn backwards, or be reflected upon themfeives in the di- rection of the chick. ‘The refleéted portion of the veffels of the veficula appears to be diftributed to the fuperior half of that bag which is now in contaé& with the external membrane. This membrane, which was in the firft periods of incuba- tion, as already mentioned, perfectly tranfparent, by its con- ne¢tion with the veficula becomes vafcular, and acquires ftrength, and contributes to form a complete fac, which en- velopes in the latter days all the humours and the chick, as will be hereafter defcribed. ~The veficula umbilicalis is found at the prefent period to have a confiderable extent, according to our obfervations much greater than has been reprefented by anatomifts in betes who do not feem to have been acquainted with the ufes of this bag in the later {tages of incubation. The chick is a very pale pink colour, or white flightly tinged with red, in confequence of the vafcularity of the furface preparatory to the formation of the fkin and feathers. When the dead of the chick is viewed potteriorly, it pre- fents the appearances of four eminences, which are an opaque white, and are joined to each other by tranfparent mem- branes. Maitre Jan confiders them as the rudiments of the bones of the cranium in a cartilaginous flate, and the tran- {parent parts as correfponding to the futures and fontanel. Haller {peaks of thefe eminences as four lobes of the brain, We conceive the account of Maitre Jan to be more correét. The head being opened at this period by Maitre Jan, he ftates that he diltinguifhed the membranes, the finufes, and the wofels of the brain, and that the fubftance of the cerebrum and cerebellum was well formed, the convolutions even being apparent. ‘This obfervation, according to our experience, is premature. ‘She ftruéture of the eye is more perfect than before. The ciliary zone, according to Haller, is as perfeét at the prefent period as at any time afterwards, Iss defcription, however, is eaftier fome days later. The Dill of the chick is cartilaginous. The animal oc- cafionally opens it in the water of the amnios as if attempt- ing to teatlow. The breait of the foctus, which was covered by a thin membrane, that in the firlt days was fo tranfparent, as to ive rife to the miltake of the heart being outfide the chelt, hasat the prefent time fome fleth upon it. About this’ period alfo the rudiments of the feraum and of the ribs may be difcovered. The gall-bladder makes its firft appearance towards the end of the eighth day, at which period, and for a few days afterwards, it is very {mall and without colour. About this time the reflections of the inner tunic of the yolk-bag, which we have deferibed at the end of the fifth day, as putting on the appearance of yellow veflels, acquire more decidedly the form of valves. They ftlll preferve the fame fituation and courfe, commencing under the bed of the foetus, and extending outwards in a radiated manner to~ wards the circumference of the vafcular area, where they terminate. ‘They are made of a number of plaits or folds which have a tranfverfe dire€tion, and which are more prominent, or have deeper folds in proportion as the val- valar lines approach the circumference of the circle. They form merely waves where they commence, and at the prefent period they are on every part of the yolk-bag much lefs marked and itriking than they become towards the end of Incubation. The Und of the 9th Day, or 216 Hours.— Maitre Jan men- tions the appearance of fome new parts at this period which we have not met with foearly. He deferibes the kidnies as fecreting a milky fluid, which he faw diftending the ureters. Te difcovered about this time the crop and the trachza and was not only able to diftinguif the principal divifions of the drain, but ftates that he could difcern the ventricles with the eminence: they contained, and even the plexus chorsides. We likewife faw upon the bafis of the brain the greater number of the nerves that arife there, more par- ticularly the optic and auditory nerves, and found the /pinal marrow to be iplit or compofed of two parts at its origin, He diffe€ted the nerves of the fpinal marrow, which, al- though very {mall, he ftates to be much more firm than thofe which take their origin from the brain. Obfery. fur la Formation de Poulet, p. 189. ; ! The furface of the foetus exhibits now a number of pores ; thefe are principally fituated upon the thighs, the back, the neck, and the wings, and are the apertures of the. cells in which the feathers are formed. The eyes are larger than the veficles of the brain; their form is not fpherical, but flattened upon the anterior and pofterior fides. The iris is ftill thin and tranfparent ; but there are feen through it and the tranfparent cornea five or fix white points which form a circle. When the eye is opened carefully with a lancet, thefe points are found to be elongated, and to form little lines, or ciliary fibres, which at this period are white. Maitre Jan ftates the eyelids to be formed at this time, but to be fo tranfparent as to be invifible, unlefs after pre- paration by a ftrong acid. The inteflines are now fo firm, that they can be examined without danger of breaking them. Vhe yolt-bag, according to Maitre Jan, begins to form two lobes, a larger and a imaller; but we have not obferved this change to take place before the 11th or 12th day. Maitre Jan made the experiment of boiling an egg at this period, after having fixed it upon a plate of lead with its {mall‘end downwards. When defpoiled of the fhell, and the chick removed, the depreflion in which the amnios Jay was fully expofed, and found to be very confiderable. He then made a fection of the egg in the vertical direction, di- viding the depreffion of the yolk through the middle, when there appeared a round {pace about the eighth of an inch in diameter, filled with+a whitifh curdy matter, and another {pace around this of a larger eXtent, containing a firmer matter than the firft, and of a paler yellow than the reft of the yolk. Maitre Jan, very erroneoufly in our opinion, fup- pofed the white curdy fubttance to be the intettines of the tectal chick. ‘Thefe do not projeét from the abdomen in {ufficient quantity to account for the appearance, admitting that they had been feparated from the chick, and had ad- D 2 hered INCUBATION. ered to the yolk during the boiling of the egg. This appearance, we believe, arifes from the fame change of orga- nization in the yolk being continued, which in the early periods of incubation produced the whitith appearance under and around the bed of the chick. The firm, pale yellow texture of the yolk which encompafles that jut mentioned, Maitre Jan accounts for by fuppofing that the fluid parts of the yolk in the neighbourhood of the chick had pafled into the amnios. There can be no doubt that different parts of the fubftance of the yolk are differently conftituted, which difference is preferved notwithftanding the motions that eggs may experience; a fact that cannot perhaps be fully explained, but which proves that animal fluids have their peculiar organization, which they can maintain as well as the folids.) When the fluids change this compofition, it is often by fome vital aétion in themfelves, initead: of a fer- mentative procefs, or the mechanical operation of the folids upon them, as is commonly fuppofed. In making the experiment of boiling eggs at this period, Maitre Jan did not find that the amnios was coagulated, al- though Haller itated that the liquid of the amnios is coagu- lable at the 198th hour. The valvular fru@ure upon the internal furface of the yo/- bag is more prominent, and when minutely examined, it is found to receive branches of the blood-veffels which are given off at right angles with the plane of the furface of the bag, and having penetrated towards the edge of each valve, there unite and form a fingle veffel, which runs within that edge. This ftrudture is better feen a few days afterwards. The End of the 10th Day, or 240 Hours. —The veficula umbilicalis has now a very confiderable extent, occupying the greatelt part of the furface of the vafcular area, which laft is ipread over almoft the whole of the yolk-bag. The portion of the bag that is applied to the white, 1s not at this date the ith of the whole. The chalazce, therefore, are much nearer to each other than when we la{t mentioned them. The white is confined almoft entirely to the lower part of the egg, and its mafs continues daily to diminith, while, on the contrary, the bulk and quantity of the yelk appears to be increafed. The external membrane is firm and flefhy where it is firencthened by the veficula umbilicalis. Some of its vef- fels are feen to extend to the white, upon which a fine pellicle maybe obferved to grow, and which is the firlt ap- pearance of a membrane to this humour, notwithtlanding fome2 authors fpeak of the membrana albuminis as an original ftructure, and others, amongft whom is B!umenbach, de- {cribe the membrane which lines the fhell as belonging to the white. The amnios is larger, its membrane more firm, and the fluid lefs clear and limpid than before. It has contracted fome adhefions with the veficula umbilicalis, and by that means, with the external membrane. Thefe adhefions are particularly in the courfe of the blood-veffels, that are diftributed to the veficula and external membrane. One of thefe vellels is always feen to pafs acrofs the back of the chick, at which place it is involved in a reflection of very fine membrane, which is connected with that of the amnios. The chick is very much curved at this period: its beak is ufually doubled down between its feet. The feathers are apparent. The external form of the bead bears a greater refemblance to that of the full-grown bird, except that the eyes are {lill enormou(ly large. ‘Lhe eyelids are evidently formed, and are occafionally ‘the vitreous humour. moved by the chick. The membrana niditans is likewife to., be feen. ward Hailer prepared at this period the ciliary zone and radiated appearance called the crown, with the membrane on which the latter is fuftained, although he {lates that thefe, parts are. formed on the 8th day, or before it. w. He fays he has feen on the 7th day the corona ciliaris, or corpus ciliare of Morgagni. At this epoch the retina, which is in thick folds, appears to go as far forwards as the cry{- talline lens, behind the corona ciliaris, with which it is co-. vered, and which is dimly feen acrofs it. He believes, not- withitanding, that the retina does not proceed to the lens 5 that the appearance of its doing fo arifes from its thicknefs and folds leaving fo {mall a {pace between it and the lens, and that the membrane which fupports the ciliary zone and crown, and is attached to the cryftalline, is different from the retina. The corona ciliaris is a row of black lines arranged alter- nately, long and fhort, in fome degree parallel, but con- verging as rays proceeding from the external circumference of the uveato the cryftalline lens. The ciliary crown is not formed entirely in the fame manner in quadrupeds and birds. In the former it appears, when examined by a micro- {eope, to be compofed of a plicated membrane, upon which fome ramified veffels advance towards the cryttalline. In birds there are only fome ferpentine lines folded upon them- felves, which are very numerous in the contour of the ciliary crown, and of which many unite to form fingle ones: they are joined together by thefe branches, and are covered with a black pigment. The figure of the ciliary zone and crown is not a perfec circle, but it is broader on one fide than the other. The greater number of anatomitts have deferibed the co- rona ciliaris as retting immediately upon the membrane of Others have believed that the retina extends to the cryttalline, and that the corona is fupported uponit. Sisce Zinn’s publication upon the anatomy of the eye, the ftructure of the ciliary body in man and quadrupeds has been well underftood ; but in birds the lamina, which ferves for the bafe of the corona ciliaris, does not come from the membrane of the vitreous humour. The membrane, which is placed under the corona ciliaris in birds, isto be feen fromthe Sth day. It is radiated, opaque, and a little afh-coloured ; more thin than the retina, and thicker than the membrane of the vitreous humour. It is of the fame dimenfions as the corona ciliaris, and in a degree attached to the corona, aad much more firmly united in the latter days of incubation; and after the exclufion of the chick, it adheres by its pelterior extremity to the great circle of the retina, and by the other to the cryftalline lens. ; The retina terminates by a diftin@ly formed circle, which is a little {trengthened at the circumference of the corona ciliaris, and at that of the zone. The plane of the retina is continued with that of the corona; the zone is placed more pofteriorly than the retina. Its external part, which refts upon the vitreous humour, is nearly horizontal, and its in-. ternal part becomes {lraight again in order to be attached to the cryftalline lens. It does not adhere to the membrane of the vitreous humour when the chick is but little advanced. Ir remains often placed upon that membrane without being attached to :t, and when the chick is more formed, or after incubation is finifhed, it quits the membrane of the vitreous humour, in order to remain conneéted to the corona ciliaris, with which it becomes infeparably united in the end. The circumference of the ciliary zone, in the latter days of incubation, can be feparated from the membrane of the vitreous * INCUBATION. vitreous humour without leaving any fhreds. It is diftin- exits from the retina by its finenefs and being feparable rom the latter without being torn ; but it is continued with the pofterior and inferior part of that.membrane. It differs {till more from the membrane of the vitreous humour ; it is thicker, it has more brown and lefs blue in its colour. Acids, when applied to it, affect its colour more than they do that of the vitreous membrane. It is fituated anteriorly upon that membrane, in another plane, and evidently fuf- tained by it, and confequently ought not to be confounded with it. The canal of Petit does not exift. ‘The membrane of the vitreous humour proceeds, without ftopping behind the ciliary zone to be attached to the cryftalline lens, that it fuftains, and which remains in its place even when the corona ciliaris has been 1aifed. The ciliary zone is eafy to prepare even in the chick but little advanced. One method is to cut off the cornea and iris; and havmmg washed away the black pigment with which the corona ciliaris is covered, it will appear between the intervals of the ciliary rays; andif fome of thofe rays be elevated, it will be feen in its proper place. It is practicable to make the lens come forth from its depreffion, and to elevate it with the corona ciliaris, with which it is conneéted ; and fometimes it may be poffible to leave the zone refling upon the mem- brane of the vitreous humour. An eafier mode of preparation is to cut the {clerotic coat through the middle, andto plunge the anterior half of the eye into vinegar. After fome hours immerfion in that fluid, if the back of a fine fcalpel be infinuated between the retina and the choroid coat, and the vitreous and cry ftalline humours be removed, one fometimes fucceeds in obtaining the ciliary zone alone, the corona remaining attached to the choroid. It is rare to have the zone entire, but a confiderable part of it is commonly preferved, aid particularly the whole of the ex- ternal circle. When the experiment fails, it isthe interior circle which remains attached to the corona, In the above account of the ciliary zone and crown, we have anticipated many facts that are obfervable during the laft days of incubation, and even after that procefs terminates, in order to render the hiftory of thofe parts more intelligible. We have alfo been induced to borrow the preceding defcrip- tion from Haller, who made the ftracture of the ciliary zone and crown in the chicka particular fubject of inveltigation, as we have not much attended to the progrefs of the eye dur- ing incubation. See Mem. furla Format. du Cocur dans le Poulet, fur )’GEil, &c. Obferv. 180, 181, 182, 191, 1925 193) 215, 216, 226, 228, 231, 239, 240, 247, 249) 2535 254,256, 261, 262, &c. and fect. xii. Mem. fecond par M. aller. The fmall tubercle, from whence the aorta and pul- monary arteries arife, aud which appears to have been a yemnant of the bulb of the aorta, entirely difappears about the prefent period. The right ventricle of the heart is ftill about the one-fourth of its length fhorter than the left. Thofe ones of the chick, according to Haller’s obferva- tions, which are now formed, are merely ina cartilaginous {tate ; even the cranium, which was apparent fome time before the prefent. The umbilical opening, through which the urachus and the blood-veffels communicate with the interior of the body of the chick, forms a projeCtion, or a fort of tube, which in afew days becomes more ftriking. Some of the,convolutions of the intellines may be feea in this opening, but the greateit art of the inteitines is contained in the body of the fetus. > The end of the 11th day, or 264 hours, the external mem- -brane goes on {lrengthening, by its connection with the ve/cu- la umbilicalis. "This laft’ is now fo extenfively united to the external membrane and the amnios, that it is only by great care, and after being macerated in fpirits, that it can be li- berated from its conneétions, and expofed as a diftin@& bag, pendent from the umbilicus. When thus feparated by dif- {e€tion, it is feen to be a large fac, compofed of thick fiefhy membrane and hardly containing any fluid. The ftrength of the bag appears to depend upon a reflection of fome of the other membranes covering and being infeparably adherent to it, for at the umbilicus the veficula is thin. Haller traced the urachus from the veficula to the re&tum after the prefent period, and even until the laft day of incubation ; but in our obfervations it has appeared that the conneétion of the vefi- cula with the chick, except by means of the blood-veffels, after this time, is very obfcure. The whole of the body of the chick is now feen to be covered with feathers. Haller at this period firft difcovered the tongue. 5 °o The fubftance of the drain is fill found to be a very fof, pulp. The /iver, in moft inftances, has a yellow colour, but in fome the red predominates. Bile is evidently fecreted: the gall-bag is green, and a greenifh or yellow fluid is commonly found in the ftomach. ‘The inteftines likewife appear to contain bile, or at leaits a bitter fluid. The renal capfules are evident ; Haller, the form of an S. The tube or fheath which projects at the umbilicus is bet- ter formed. The blood-vefels, which come out at this place, are eafily traced from ditterent fources. Thofe which are diltributed to the external membrane and veficula come from the loins, and thofe belonging to the yolk-bag are from the abdomen, The end of the 12th day, or 288 hours, the chick has more of the figure of the adult bird than at any preceding period. ‘The extremities are larger, and the head {maller, in preportion to the reftof the body. The bill is more pointed, and of a harder fubitance. The fomach has gained greater ftrength ; it contains a white coagulated or curd-like fubftance. _There is dile found in all the inteftines, which is very bitter. On the Outfide of the umbilical fheath there are fome folds of the inteitines, with one of which the yolk-bag is feen to be connected by a thort procefs, which is o-denuy we continuation of the alimentary canal into the yolk-bag. This part has commonly, and as we fhall hereafter fee very defervedly, received the name of du@us witello-inte/tinalis. Al- though we have not obferved the duct before the prefent date, there is every reafon to fuppofe that it is coeval with the inteltines. Ma'pighi ftates that he faw the ffomach of the chick on the outlide of the umbilical fheath’; but this would feem to be anerror. The ftomach, it is true, is fituated low down in the belly, but, as far as our obfervation goes, it never is protruded from that cavity. The ttructure of the eye is the fame that is deferibed on the 11th day. It is proper to remark, that at no period is there to be found in the chick a membrana pupillaris. The yolk-bag, according to our obfervations, begins in general to alter its figure. Before this date it was an irregularly round mafs, containing two depreflions; one for accommo- dating the chick and amnios at the upper fide, and another at thie lower part of the yolk, in which the white is partially lodged. The latter is a regular deprefiion, and before this period contracts in breadth, altHough it becomes fomewhat greater they have, according to INCUBATION. greater in depth; an alteration which arifes from the:diminu- tion of the white, and from the chalaze tending to ap- proach each other, and thereby to bring the two poles of the yolk, at which thefe bodies were originally attached, to- gether on the lower furface of the yolk. The approxima- tion of the chalaze feems likewife to occafion, in the firlt in{lance, the divifton of the yolk-bag into lobes. At the prefent period the bag forms two lobes, one large and one fmall. ‘Thefe afterwards, by the compreffion of the chick, are ufually moulded into three, but in the egg of fome other birds into a greater number: thus the yolk-bag in the Guinea fowl! is divided into fix or feven lobes. The End of the 13th Day, or 312 Hours.—Upon opening an egg at this period, and removing the contents from the fhell, the qd&ite is found to have affumed a greenifh hue, and to have more confiftence than at any of the former periods ; and it is inclofed entirely by a fine film, which retains it in one pofition, and in flrict connection with the depreffion on the inferior furface of the yolk-bag. The external veffels are diftributed over the whole white. The yolk-bag has {till more of the lobulated figure, and the chick lies obliquely acrofs the bag in the fuperior de- preffion between the two lobes. Vhe valvular ftru€@ture on the internal furface of the yolk-bag, which we have noticed before, has now arrived at perfection. The valves are more eminent, form numerous folds or coils, their edges appear to be covered by tubes which are convoluted upon themfelves in the manner of anteflines. Thefe have been called by Haller the vermi- form and inte/finiform tubes, and have been deferibed both by him and other anatomitts as real veffels: but they have al- ways appeared, in our examinations, as reflections or pro- cefles of the valvular ftru€ture, in the fubftance of which the real blood-veffels ramify. Maitre Jan fpeaks of them as veffels which colle& the fluid of the yolk, and which afterwards terminate in the blood-veffels. Haller confiders the blood-veffels which run in the edge of the valves as a diftin® order of veflels, that end in the trunks of the veins of the yolk-bag, and which take their origin at the circle, that exilts in the latter days of incubation around the line where the yolk and white are joined to each other. We have already faid that the veflels upon the edges of the valves have numerous communications with thofe that ran along the roots or bafes of the valves, and which belong to the external tunic of the yolk-baz, and were originally the ; “lm that formed the vafcular area. ‘The venous circle of the yolk-bag, we have no doubt, is the fame that in the early ftages of incubation has been defcribed as the contour of the vafcular area: and the vermiform tubes appear to be the villous or fpongy procefles of the external furface of the yolk-bag, which appeared inthe firft days greatly extended, and grown into the form now defcribed. The chick, at the prefent period, has ftill more of its perfeQ fortn ; its head is much curved, and the billis placed under the, wing, which polition is preferved throughout in- cubation. The dil/ is pointed and has its natural figure. Haller tates, that the chick is often perceived to open and fhut the bill about this period and afterwards. Maitre Jan, on the contrary, although the chick moves its limbs frequently and itrongly, afflerts, that it does not open its mouth. We have not attended to this circumftance, from confidering it of little confequence, and no way explaining the mode in which the chick is nourifhed, The internal organs of the ehick are confiderably advanced towards perfection. The drain, although {till foft and pulpy, is well formed. The ddnies and-tefes are completed in their thease figure. The /pleen refernbles a conglobate gland. The lungy are of greater fize than the heart. ‘ a: The firft branch that the aorta gives after its exit from the heart is the /eft carotid, which gives off the fubclavian ar- tery of that fide. The trunk of the aorta then forms an arch, which turns to the right. It fends off the right carotid, and approaches the vertebre im palling behind the right lung: it is umited to the ductus arteriofus above the liver. : The right ventricle of the heart is till fhorter than the left y it is of an oval figure, but the latter ends in a pomt. Lhe end of the 14th day, or 336 hours, the changes inthe humours and the formation of the organs of the chick are found to have made progrefs fince the preceding day. We have already ttated that the membrane which is ex- ternal to. the chick and the yolk-bag, and which is fearcely palpable in the firft days of incubation, gains ftrength and receives veffels after the velicula umbilicalis bas acquired fome fize, and has adhered to it. ‘The magnitude of th® veffels, and the thicknefs of this membrane, increafe in proportion ag the veficula umbilicalis grows: we therefore conceive that the developement of this membrane depends upon its connection with the veficula and the blood-veffels which it derives from thatbag. We have obferved, that fome days before the prefent period, the external membrane begins toextend over the white, which it ufually enclofes about the 14th day: it therefore forms a perfect fac, containing the chick in its amnios, the yolk-bag, and the white. The veficula umbilicalis about this period has arrived ata very great fize; its extent cannot be ealily afcertained, as it is fo much incorporated with the external membrane, that they cannot be feparated or accurately dittinguifhed: but we fhould conceive that the veficula exceeds in fize the yolk-bag, as there is the appearance of the external membrane being thickened and {trengthened by one under it as far as the yolk extends. The manner in which the veficula concurs to form a mem-~ brane which invetts all the contents of the egg, appears to be fimilar to the reflection of the ferous membranes of the thorax. and abdomen, inorder to cover the vifcera and line the parietes of thefe cavities. It fhould however be obferved, that when it is opened, there is no appearance of its forming an imme- diate tunic fimilar to the pleura pulmonalis, but it feems to be a fimple bag, holding the contents of the egg, and only united to them by fome reflections of very fine membrane, which are chiefly attached to the back of the aninios, and in the courfe of the trunks of the blood-veffels. This circum- {tance has embarraffed other obfervers, and has led man to overlook the fhare that the veticula umbilicalis has in the formation of the common membrane of the parts of the egg. The blood-veffels which are diltributed to the external mems brane are, even at the prefent period, much larger than thofe of the yolk-bag. There are four long branches, which fend out numerous ramifications, that become fo minute as to evade the fight; but when this membrane is viewed with a mag- nifying glafs, it prefents a beautiful fpe€tacle. The fituation of the four great branches is not conttantly the fame ; as they are difperfed upon a loofe membrane, which varies ia fome degree its pofition according to circumftances: one of them, if not two, is always feen, however, to crofs the back of the ammnios. The four branches are arteries, and are formed immediately on the outfide of the umbilicus of the chick from the two iliac arteries; they, therefore, correfpond to the umbilical arteries of mammiferous animals. The left iliac artery is always of much greater fize than the right; it even appears larger than the aorta, of which it is the termi nation. INCUBATION: nation. The right iliac artery is often a mere filament, and in fome inftances it feems to be wanting altogether. The venous. branches which accompany the umbilical ar- teries are fo {mall that they are hardly diftinguifhable, al- though the former are of fufficient magnitude to be injected, and have a vifible pulfation. There is, therefore, a quantity of blood fent to this membrane which is not returned to the foetus. The colour of the blood, both in the umbilical ar- teries and veins, is a purple. The vafcular membrane, above defcribed, has received different names. By the older anatomi{ts it was ufually called the chorion, to which perhaps it is moft analogous. Haller gave it the name of the umbilical membrane, and Le- veille has lately called it the bag-haped membrane (membrane Jfacciforme), from the circumftance of its enveloping all the parts of theege. We have hitherto called it the external membranc, until it fhould be defcribed; but Hereafter we fhall fpeak of it as the chorion. In the account of the unin- cubated egg, the membrane, on which the fhell is moulded, is called the external, which it ftri@tly is. It fhould not be con- founded with the chorion, from which it differs fo much ia figure, firu€ture, and ufe. See Ecc. 5 Haller appears to have not diftinguifhed the chorion from the membrane of the yolk-bag, which contains the vafcular area. “The progrefs of thefe two membranes is exceedingly different, and the fyltem of veffels belonging to each is perfeGtly diftinét. The valvular area is confined to the furface of the yolk-bag, and is intimately connected with its membranes, and the valvular apparatus already defcribed. ‘The veffels of the area are chiefly venous, and are formed independently of the heart of the chick. They are moft numerous in the ear'y periods of incubation, although they increafe in fize along with the other parts of the chick. They go to terminate in a large vein which proceeds direétly to the auricle of the heart in the firit days, but, after the liver appears, is feen to pafs behind it, and when the inteftines are formed is found upen the mefentery, and conttitutes the largeft branch of the vena porte. The arteries of the vafcular area are infignificant, uatil towards the end of incubation. When firlt perceived, they are given off from the dorfal aorta, and after the develope- ment of the inteftines they are difcovered to be branches of the mefenteric artery. The veffels diftributed to the area appear, therefore, analogous to the omphalo-mefenteric arteries of quadrupeds, although their deltination and func- tions may differ. The weffels of the chorian, as we have already faid, are branches, or rather, in the foetal flate, the trunks of the iliacs extended uponthis membrane. They do not exilt pre- vious to the appearance of the veficula umbilicalis, with the growth of which they keep pace. The arteries of the chorion @xceed the fize of its veins in the fame proportion as the veins of the yolk-bag exceed the arteries of that part. The vafcular area, therefore, collefts and tran{mits the b'ood to the foetus, while the arteries of the chorion difpofe of a great portion of it in an extraneous circulation, and in the formation ofa thick and extenfive membrane. Haller and others have defcribed, at this date, an appear- ance of white veflels upon the albumen. Maitre Jan reckons feven of thefe white lines or veflels. ‘Chey avife from a little circle which is in the centre of the white ; they divide into branches in advancing towards the vafcular circle of the yolk- bag, which forms at this time the boundary of the feptum between the yolk and the white. Maitre Jan defcribes them as pafliag beyond the feptum and extending to the yolk. Haller conjectures that the ufe of thefe veilels may be to abforb the white and convey it into the yolk. Obfer,. 226. Mem. 1. ‘We have not made fuch an examination of thefe white lines, as to enable ns to pronounce whether they are veffels or not, or what their ufe may be. The eater of the amnios appears to diminifh about this period of incubation. It alfo becomes gradually lefs pure. It is ftated by Maitre Jan, that when the fluid of the am- nios is boiled at this date, the greater portion of it acquires the hardnefs that the white of the egg poflefles after being boiled, although previous to the experiment it is much more fluid than the albumen ; from whence Maitre Jan conjeétures that fome of the fluid of the white is conveyed into the fac of the amnios, but he does not pretend to explain the means. by which this is effected. The curd-like fubftance, already mentioned as being found in the ftomach and inteftines, 1s now met with alfo in the crop and cefophagus. According to Haller, boiling water makes this fubftance hard in the fame manner as it does the white of an egg. The end of 15 days, or 360 hours, the membrane of the chorion continues to increafe in iirength, and its arteries in fize. Haller fpeaks of having fometimes found the latter of a lighter red than the veins, which does not accord with our obfervations. We have generally perceived the arteries and veins to be fimilar in colour, or at leaft when a difference did exiit it was immaterial, and fometimes the arterics, and fometimes the veins were the lighter coloured veflels. ‘The vef- fels of the chorion have always appeared darker in our obfer- vations than thofe of the yoik-bag, but the charaer of all the blood is venous in the incubated egg during the latter periods. The reflections of membrane which accompany the principal veflels of the chorion, are now more firm than »efore, and continuous with them appears a fine tranfparent membrane that envelopes the yolk as far as the vafcular circle on the edge of the white, and extends over part of the amnios, to which membrane it adheres, fo as to cover an irregular triangular portion of the chick in its amnivs, and inclofe more loofely the umbilical fheath with the convolus tions of the inteflines which proje&-from it and the neck of the yolk-bag. This membrane does not appear to have been. well underftood by the older anatomitts. They fometimes {peak of it asa layer of the yolk-bagy and at others as the veficula umbilicalis, or allantois. The bett defcription of it kas been given by Leveiile, who has alfo- named it very appropriately entero-clorilyme, or the membrane containing both inteflines and yolk. ‘Lhe origin of this membrane is {tillinvolved in a degree of obfenrity. Leveille deferibed it when perfectly formed, but has thrown no light upon the hiftory of its growth. The: prefent date isthe firit at which it has appeared to us to be perfected, but we conceive that it is one of the oriyinal membranes and the fecond layer, that in the firft days comes off the vafcular area as a very fine pellicle. It feems to be conneéted, or rather continued into the peritoneum, lining the abdomen in tiie late periods of incubation, of which we fhall {ee farther proofs hereafter... There is no reafon for fup- poling this membrane to be the veficula umbilicalis, or allan- tois, as the veficula is a {tronger membrane in the middle of incubation than this membrane is feveral days. afterwards, when it is completely formed The proper contents of the allantois alfo are found, throughout incubation, within the cavity of the chorion, to the Forstnts of which we have fuppofed that bag contributes, but never under the membrane that inclofes-the inteftine and yolk-bag. Haller ftates that he examined the valves upon the inner furface of the yolk-beg with.a lens, aad diftincly faw the inteftiniform INCUBATION. mteftiniform tubes upon the edges of the valves moft beauti- fully curled and folded upon themfelves. He failed to fee the aperture of thefetubes. By maceration in water they were detached from the valves, and fwam in the water. The valves themfelves, by a longer maceration, were alfo fepa- rated, ; The chick increafes regularly in fize, though not fo rapidly as during the early part of incubation. Haller obferved the feathers upon the eye-/ids at this period, and that the /pleen had acquired the figure it preferves through life. The curdy, or coagulated fub/fance is now found in the cefo- phagus and the crop, as well as in the ftomach and inteftines. Harvey found this fubitance principally in the ftomach : this appears to be the cafe, and when met with in other places, it would feem to have proceeded from the ftomach to them. The end of 16 days, or 384 hours, the fpace between the two layers of the membrane lining the fhell, for contain- ing air, 1s much increafed fince the period at which we have tal mentioned it. s\bout one-fixth of the whole cavity of the egg is occupied by air. &» & reflections of fine membrane which attach the chorion to the other membranes ; 4, /, the two trunks of the umbi- lical arteries, which each divide into two branches, and are diftributed to the chorion. " Fig. 17. is intended to explain the formation of that membrane which includes the yolk-bag and the inteftines of the chick, and for this purpofe the chorion is intirely removed ; «a is the chick contained in the amnios; 4 indi- cates the membrane of the yolk-bag and inteftines ; it is feen to cover a certain extent of the amnios and chick; it defeends pofteriorly to be conneéted to the feptum, be- tween the yolk-bag and the white: anteriorly it is laid open, in order to turn out the yolk-bag, which was before contained in this membrane. Within the chink made by the incifion are to be feen the inteftines of the chick perfe@ly uncovered and defignated by c. The yolk-bag hangs by the ductus vitello inteftinalis and its blood-veflels, and is marked by the letter d. Fig. 18. fhews an egg with the great end fra€tured pre- vious to the exclufion of the chick: fome more pieces of the fhell are removed in order to bring into view the membranes that are lacerated by the bill of the chick 3; ais that portion of the membrane under the fhell which forms the air-cell, and in which there is a fmall aperture. “The chorion and amnios are torn to a greater extent to permit the bill of the chick to appear on the outfide of the fhell ; b the bill of the chick: befide it aré feen fome points of fea- thers proje€ting out of the opening made in the outer membrane. Fig. 19. isa portion of the fmall inteftine taken from the adult bird, on which the appendix is placed, that we have deferibed as the remnant of the du€tus vitello inteftinalis ; a the inteftine; 4 the appendix or cecum; c is a little knot which was, during fcetal life, the yolk-bag. In fome of the preceding figures, the form of the chick is reprefented at different ages, which the reader will under- {tand without the letters of reference. Of the Ufes of the different Parts of the Egg during Incuba- tion, and after the Exclufion of the Chick.—Although there ean be no doubt that the chick derives its nutriment from the humours of the egg, the mode in which this is effected is involved in confiderable obfcurity. Anatomifts have entertained very different opinions upon the fubject. Some have fuppofed that “both im mammiferous animals and birds, the foetus is nourifhed by {wallowing or abforbing the liquor amnit. Heiter, in his Compendium of Anatomy, ftates, that after the contents of the uterus of the cow were frozen, he found a piece INCUBATION. a piece of ice of the thicknefs of a finger extended from the congealed amnios into the mouth of the foetus, and down the cefophagus as far as the ftomach. There have been fometimes feen alfo in the ftomach of the fetal calf the fame kind of fatty concretions, that ufu- ally exift in the amnios of this animal, The hairs of the calf have been difcovered in the meconium. New-born in- fants have been obferved to vomit a fluid refembling the liquor amnii. It has been remarked by Harvey, Haller, and feveral other writers upon incubation, that the foetal chick frequently opens and fhuts its mouth while inclofed in the amnios, as if it were performing the act of {wallowing. The preceding facts leave no doubt that the water of the amnios paffes into the flomach of the foetus, but by no means proves that it contributes to the nourifhment of the young animal, but rather the contrary, as if it were either digefted in the ftomach, or confumed by abforption, it could not be found towards the end of gettation, or after- birth, in the quantity reported. Thofe who give credit to the nutrition of the foetus by the liquor amnii, cite fome remarkable cafes in which the umbilical cord is faid to have been deftroyed before birth, or in which the infant has been born with the cicatrix at the umbilicus perfectly formed. See Obferv, rarior cent poft pars 1. obf. 32.—Journal des favans, année 1673, p- 69. Obfervat. communiq. par Chatton.—F. Hoffmann & Triller Differt. de pinguid e fucco nutr. fuperfl. p. 10. and Rommel Ephemer. nat. cur. dic. 2 2 7, obf. 209. It fhould be remarked, that of thefe obfervations, fome are related in a manner which makes their accuracy very doubtful; and others, even if corre&tly ftated, do not prove the point they are propofed to eftablifh. Thus the infant with the cicatrized navel was not feen by Chat- ton until five months after birth; and in fome of the cafes where the cord was faid to be injured or putrid while the child was alive, it does not appear that the circulation had been deftroyed. We fhould not have noticed the doGtrine of the nutrition of the foetus from the liquor amnii, as it is generally dif- sredited, had it not been advanced with fome limitation by Lobftein in a modern publication. This author does not pretend that the foetus {wallows the liquor amnii, he allows that the young animal exifts and increafes in growth, before the digeftive organs or even the mouth are formed. He is alfo aware, that'there have been montters produced in whom the head was wanting. He therefore fuppofes, that the fluid of the amnios is taken into the body of the foetus by the abforbents of the fin, which opinion he grounds upon the following circumftances : Many of the cold-blooded ani- mals, he fays, have no placenta, but in every {pecies the fcetus floats in aclear liquor, analogous to that of the amnios. This fluid, he remarks, diminifhes in proportion to the growth of the foetus ; it likewife remains uncorrupted during foetal life, which faét, he thinks, can only be explained by its pafling fucceflively into the body of the young animal. (Lobitein fur la Nutrition du Fetus, § 88.) Our reply to this reafoning would be, that the liquor amnii is not a nu- tritive fluid perhaps in any inftance ; that when firft perceived, it is in a very {mall quantity: it is,in the higher clafles of animals, contained in a membrane which is continuous with the fkin of the foetus, and which membrane further derives its blood-veflels from the fetus. It is, therefore, a fecretion of the young animal. During feveral days of incubation it increafes as the chick grows, and in the latter periods does not perhaps actually diminifh in quantity, but ceafes to in- oveafe in proportion to the developement of the chick ; for which reafon it appeats lefs than it did when the chick waa {mall. The quantity of the liquor amniiat any given time, — even admitting that it is not fecreted by the foetus, would be quite infufficient for the formation of the latter. The principal part of this fluid alfo is loft when the membrane that contains it is ruptured, previous to the liberation of the young anima! from the uterus orthe egg. We fhould, on all thefe grounds, deny that the liquor amnii is deftined to fuftain the fectus in any circumftances. In all oviparous animals, there is no neceflity for fuppofing wT the liquor amnii to be a nutritive fluid, as there are other humours provided for the purpofe of nourifhing the foetus. — The fources of nutriment, during foetal life, are particularly - evident in the ovum of birds. In the hiftory of incubation it will be found, that the albumen begins to diminifh as foon as the feetus becomes ap- parent, that it continues to decreafe regularly as the chick grows, and that this fluid is entirely confumed at the end of mecubation: there is, therefore, no doubt that the white of the egg is the fubftance from which the chick is formed, It has been fuppofed by fome writers, that the yolk alfo contributes to the fupport of the chick while in the egg. This opinion refts chiefly upon the circumftance of the yol becoming larger and more fluid after the white begins to . diminifh, which has been attributed to the albumen pafling into it, that the two fluids might be mixed together, and afterwards converted into the fubftance of the chick. Mai- tre Jan, Leveille, and others believed, that the white is ab- forbed by the chalaze, and thus conveyed into the yolk-bag, and Leveille has even pretended to demontlrate the tubular ~ ftruGture of one of the chalazez. We have already ftated; that by the ftri¢teft examination we at not perceive the ftru€ture defcribed by Leveille ; we are therefore led to con- clude, that the fluidity of the yolk, during incubation, is oc-_ cafioned by the organization of that humour undergoing a — change from the application of heat. We have found alfo the yolk become fiuidywhen eggs were fat upon, which had not been fecundated by the cock, and in which the white was not fenfibly diminifhed. The weight of the yolk at the end of incubation, almoft proves that it is not concerned in the nourifhment of the chick, until after the animal is excluded from the egg. P We confefs, however, that we are unable to form a decided opinion with refpeé& to the manner in which the albumen is carried into the body of the chick. ‘Throughout incuba- tion the white tends to the lower and {mall end of the egg ; the chick is placed at the upper and great end. There 1s, therefore, no direét communication «between the chick and the albumen. The blood-veffels of the egg have apparently no immediate conneétion with the white, until its proper membrane is formed, at which time this humour is already much reduced, and even then there are only a few wander- ° ing veflels fpread over its membrane. ‘Thefe faéts afford reafon for the fuppofition that the white paffes into the yolk- bag, through the chalaze ; but, on the other hand, it is fcarcely credible that the white, after being mixed with the yolk, fhould alone be abforbed during incubation, and the latter fluid left undiminifhed. The chalaze, whether they perform the office of abfor- bents or not, act as ligaments to the yolk-bag, and have the effect of keeping, throughout incubation, that furface of the bag uppermoft on which the chick is fituated, in order that it may receive heat from the hen. See Ecc. The ufe of the veficula umbilicalis is a fubje@t on which no fatisfactory theory has been yet made out. ‘The older anatomiits fuppofed this part to be a refervoir of urine, and gave it the name of allantois, Lobftein confi ders ¥ INCUBATION. ders the fluid of the veficula and the allantois as the nutri- ‘ment of the foetus during the firft periods of its exiftence. Soemmerring and Blumenbachfuppofe that the veficula umbili- calis is intended for the nourifhment of -the foetus in the hu- man fubjeét. Although it does not appear that this ufe of the veficula or allantois has been in any degree proved, the argu- ments employed are conclufive againft it being a refervoir for urine, and particularly fo with refpeét to birds and the human fubjecé. 7 The veficula umbilicalis attains a confiderable fize, and is filled with its proper fluid before the kidnies exit; and although the membrane continues to increafe, the quantity of the fluid it contains by no means keeps pace with the growth of the kidnies. We cannot admit the reafoning of Haller, who concludes that the exiftence of fluid in the allantois is a proof that the kidnies fecrete urine before they are vifi- ble. It has not yet been afcertained that the fluid contained in the allantois is urine even in quadrupeds, where this bag communicates fo plainly with the bladder. Lobftein found, after the allantois had acquired a very confiderable bulk, that the kidnies were {mall and pale, and refembled organs which had not yet been called into action. The bladder alfo was contracted. He very judicioully remarks, that if the aljan- tois were diftended to the fize that it is known to poffefs in the early periods of geftation and incubation by the fecre- tion of the kidnies, it would become before birth a molt prodigious fac. It is extremely improbable that the fecre- tion of urine fhould diminifh or be fufpended after the kid- nies were perfe@tly formed, by which fuppofition alone the {maller relative fize of the allantois in the latter periods of foetal life could be explained We may further obferve, that there is no refervoir provided for any of the other fecretions of the foetus, and that even the gall-bag is not remarkably diftended, although it evi- dently contains bile fome time before birth, and notwith- ftanding the liver is the largett gland in the body. In deferibing the progrefs of incubation, we have mention- ed, that the veficula umbilicalis of birds contributes to the formation of the membrane called the chorion ; that its fluid difappears, leaving behind it fome layers of a foft cal- careous fubftance, refembling the excrement of birds, or rather the folid parts of the urine of thofe animals. This excrementitious matter has been confidered a proof of the veficula being the receptacle of the urine, but it fhould be obferved that the fame fort of fubftance is found after the confumption of all the fluids of the egg. The funétions of the veficula umbilicalis in birds, if there be any peculiar to it, are loft in thofe of the chorion. By recurring to the hiltory of incubation, it will be found that the external membrane of the yolk-bag, on which the vafeular area is fpread, and the chorion, receive a vattly larger quantity of blood than circulates at any one moment in the body of the chick. The chorion is not originally a vaf- cular membrane ; its veffels arife with the veficula umbilicalis, by which they are conducted froni the body of the fotus. ‘The vafcularity of the chorion in fome degree fucceeds that of the vafcular area; it appertains chiefly to the arterial fyltem, whillt that of the area is almoft entirely produced by veins, The ufe moft commonly afcribed to thefe two great fy{ftems of extraneous vellels, is the oxydation of the blood of the fcetus, for which purpofe it is alfo believed that the air is ad- mitted into the great end.of the egg. We have, however, been led to entertain a different opinion of the offices of the vafeular area, chorion, and air-cell, in confequence of fome experiments we have made, which, we hall lay before the reader, Experiment 1.—Two eggs of a hea weré opened at the great end by deftroying the fhell and outer layer of the membrane compofing the air-ceil. The blood-veffels were feen through the membrane of a deep purple colour. Being expofed to the atmofpheric air, they became, in a few minutes, a vivid fcarlet. "The fame eggs were then placed in a jar filled with fixed air, and their blood very foon re- affumed its original purple colour ; and upon the eggs being again brought into the common air, the {carlet colour of the blood was fpeedily revived. One of thefe eggs had its blood rendered venous and the other arterial, in the manner above-mentioned. Both were then placed in a veffel of water, in which they remained all night, and on the next morning each continued to preferve its proper colour. Ex. 2.—We took an egg which had been fat upon 18 days, and having removed a fquare portion of the fhell and membrane forming the air-cell, about a quarter of an inch in fize, it was placed in a jar of carbonic acid gas, over a bath of quickfilver, which was kept throughout the experiment, as nearly as poffible, at the temperature of 104° Fahrenheit. The motions of the chick caufe an egg to roll upon a {mooth furface, but in this inftance no motion was perceived. After the egg had remained an hour in the fixed air it was taken out, and the aperture of the fhell was carefully clofed with » paper and glue, and two hours elapfed before the egg began to roll upon the furface of the mercury. Upon this occa- fion, therefore, the chick appeared to have been fubmitted three hours to the influence of carbonic acid gas without - being deftroyed. Ex. 3.—An egg, at the 18th day of incubation, was © deprived of the greateft part of the fhell, and of the mem- - brane that lines the fhell, fo that the veffels of the chorion were fully expofed. The membrane of the chorion was moiftened to increafe the effet. The egg was fuffered to remain for 25 minutes in the fixed air, during which time the chick fhewed no motion; but upon being removed into atmofpheric air it ftruggled as ufual. Ex. 4.—The air contained in the cells, during the latter ftages of incubation, was collected from a number of eggs, and fubjeéted to experiment in a graduated glafs tube. It afforded, as nearly as may be, the fame produéts that are obtained from atmofpheric air, after it has been f{poiled by refpiration ; from which it would appear, that it was the prefence of undiluted fixed air in the 2d and 3d experi- ments that caufed the motions of the chick to ceafe, and not the abfence of oxygen gas.: Ex. 5.—Being aware of the fatal effe&ts upon the chick, which have been reported -to follow covering the fhell of the egg, fo as to ftop the further admiffion of air into it, we varnifhed the furface of feveral eggs at different periods of incubation. The refult of this experiment was nearly the fame in every inftance. The chick always died ; not, however, immediately, but in two, three, or four days, and apparently not in confequence of being unfupplied with frefh air, but from being unprovided with any kind of air to occupy the fpace caufed by the expenditure of the fluid contents of the egg in the formation of the chick: for, in thefe cafes, we found the yolk more or lefs diforganized, the membranes fometimes ruptured, and the foetus always mal- formed or imperfectly developed. If the death of the chick had been occafioned by the deprivation of oxygen, it would have been immediate, and unattended with any change in the ftructure of the fertus, The graduabadmiffion of air into the eggs of birds takes place as a matter of neceflity, becaufe the fhell, being hard and unyielding, cannot» accommodate its form to the bulk of its contents in proportion as they diminith ; for the parts orig nally INCUBATION. nally contained in the egg occupy a much larger f{pace, as already mentioned, than when the chick is fully formed. The preceding experiments prove that the air does not pafs into the egg for the purpote of effe&ting any change in the blood of the fcetal chick, fimilar to the procefs of refpi- ration, as has been generally fuppofed. It is true, never- thelefs, that when the blood is firlt expofed to the air con- tained in the great end of the egg, it becomes oxydated by it; and hence, during the firit days of incubation, all the blood-veffels of the vafcular area are obferved to be of an arterial colour: but as there are no means of expelling the air as it becomes f{poiled, and as the fupply of pure air after- wards is too {mall in quantity to produce any effect upon the blood, all the veffels are of thé purple or venous colour in the latter periods of incubation. This appearance of the blood has been obferved to exift by many writers upon in- cubation, and coincides with the experiments we have re- lated above. It may be afked, what end is anfwered by the diftribution of fo much blood to the membranes of the egg, if it be not for its expofure to oxygen gas? This is a queftion which, perhaps, cannot be folved by dire€t experiments. We conceive, however, that itis neceflary, during foetal life, to have a greater quantity of blood formed than is em- ployed in the organs of the young animal. The functions of the blood-veffels of the foetus are almoft confined to the creation of its body ; but in the adult animal the principal part of the mafs of blood is confumed in furnifhing the va- rious fecretions. The vafcular membranes of the ovum may be confidered as performing the office of the various fecretory organs, the actions of which are dormant, at leaft during the early periods of fcetal life. By this means alfo the or- gans of the body are accuftomed to the preparation and prefence of a quantity of blood which is wanted immediately after the young animal leaves the uterus or the egg: indeed, without fuch a provifion, it would feem to be impoffible for the foetus to affume, in a moment, the func- tions of an independent {tate of exiftence. Further, the extraneous circulation on the membranes ap- pears to be the means by which the nutritive fluids of the egg are converted into blood. This is almoft demonftrated in the vafcular area, in which the red blood appears before its veflels have any vilible conneétion with the body of the chick. . The membrane containing the vafcular area, therefore, we deem the organ of nutriment and affimilation. It corre- {ponds to the different vifcera which convert the food into blood in the full-grown or perfe&t animal. The chorion of birds fupplies the place of the vifcera, which return the particles of the adult body to the common {tock of nature. We could draw many arguments in fupport of this theory from the anatomy of the lower claffes of animals, and from the ftructure of the vafcular fyftem in particular animals of a higher rank ; but it’ does not feem confiftent, with the nature of thé prefent article, to enter into an extenfive phy- fiological difcuffion. If it be admitted that the blood of the foetus does not re- ceive that effect from the air which gives the arterial cha- racter, it becomes defirable to afcertain the power of the feetus for generating animal heat. With this view we have inftituted the following experiments. Lx. 1.—An egg, on which the hen had fat 17 days, was placed upon a bath of mercury heated to 104° of Fahrenheit, and kept at that temperature for feveral hours. ‘The heat of the room was 75°. The upper furface of the membranes, when the fhell was broken to admit the thermometer, raifed he inftrument to 92°. 'T’he interior of the egg was 95°, and on the inftrument being thruft down fo as to touch that part of the fhellin conta& with the mercury, it did not rife highe® than 100°. From this experiment, the ftandard heat of the chick appeared to be about 95°. ; Ex. 2.—An egg, at the fame period of incubation as the la{t, had the fhell entirely removed, and was placed upon a mercurial bath at the heat of 83°, the air of the apartment being 75°. The centre of the egg preferved the anftrument at 88°, and when pufhed down upon the membranes next the mercury, it did not fail below 85°. 2 Ex.3.—A chick, about 18 days old, was removed from the egg and its membranes, and expofed for fome time upon the furface of a bath of quickfilver at the temperature of 79’, and becoming gradually cooler as the heat of the fur- rounding air was 75°. The thermometer was introduced into the thorax of the chick, and flood at 80° for a fhort » time, even after the chick had died. The heat of the different parts of the incubated egg can= not be expected to correfpond with each other-as long as external heat or cold is applied to one furface only, but the three laft-mentioned experiments, notwithftanding, appear to us conclufive, with refpeét to the foetal chick pof- {effing an independent temperature, and being capable of refifting, to a certain extent, both foreign heat and cold. It is proper to obferve, that we have made fome experi- ments with refpe&t to the oxydation of the blood, and the animal heat of the foetus in mammalia, which entirely cor- refpond in their refults with thofe above related ; and, there- fore, fortify our opinion with regard to the abfence of any procefs analogous to refpiration during fetal exiftence, and likewife prove that the young animal has an independent temperature. We have already mentioned that the yolk lofes very little, if any, of its original weight during incubation ; but that after the chick is excluded from the fhell it is. fpeedily confumed. This humour, therefore, is obvioufly defigned for a provifional nutriment to fuftain the chick until it is able to procure a fufficient quantity of its natural: food. This difpofition of the yolk is particularly necef-: fary to thofe birds that live in a wild ftate, but is ufeful. to all. MAT, Anatomifts are divided with refpe& to the manner in: which the yolk is conveyed into the fyftem of the chick. - Some affert that it paffes into the alimentary canal, through’ the duus vitello inteftinalis, where it is digefted previous to its being abforbed by the laéteals. Others deny that it is to be found in the ftomach or inteftines, and believe that it is abforbed immediately from the yolk-bag. We are. difpofed to adopt the latter opinion, both from never having being able to difcover any yolk in the inteftinal canal, and from a number of reafons, which indirectly go to difprove the paflage of the yolk through the duétus vitello intefti- nalis. In the firft place, there does not feem to be any power by which the yolk can be tranfmitted into the intettine. The yolk-bag itfelf has no mufcular coat, and the mere . preflure of the parietes of the abdomen of the chick, even : if fufficiently ftrong, would be exerted as much upon the inteftines as the yolk-bag, and might therefore as readily. urge the contents of the former into the latter, as the yolk, into the gut; for the ductus vitello inteftinalis does not penetrate the coats of the inteftine obliquely, fo as to pro- duce the effeét of a valve, but has merely a flight contraction at the orifice, which mutt operate equally on both fides. There would appear to be no neceflity for the yolk to pafs into the alimentary canal, or to be fubjected to any procefs of digeftion or aflimilation previous to its entering - the <— = Inc Whe fyftem of the chick. It is already a vital fluid, which ‘is proved by Mr. Hnnter’s experiments upon the tempera- ture of eggs. (See Ecc.) If the humours of the egg did not pofleis vital properties, they would very foon become putrid from the heat of incubation. When we contemplate the ftructure of the yolk-bag, it muft be perceived to be not only continuous with the intef- tinal canal, but to refemble the latter in organization, ex- cept that the mufcular coat is wanting. The external membrane of the bag differs in no refpeét from the peritoneal coat of the inteftines; and the internal membrane is fo analogous to the villous coat, that we*cannot hefitate to al- low them the fame funétions. No advantage, therefare, would be gained by the paflage of the yolk into the in- teftine. The: abforption of this humour, by the valvular or ‘in- ternal membrane of the yolk-bag, feems proved by the organization of the yolk undergoing a change upon the furface which’ is in. contact with the membrane, while the central part is unaltered. he circumftance alfo, of a con- crete fubftance, refembling the refidue of the albumen, being left in the yolk-bag after the nutritious parts of the yolk are confumed, affords a flrong evidence that the latter are removed by a procefs of abforption ; for if the yolk pafled through the duétus vitello inteflinalis, it would difappear without diflinGtion of parts. The authors who believe that the yolk is carried into the fyftem without a previous preparation in the alimentary canal, attribute the office of abforption to the red veins. In proof of this opinion Blumenbach and others flate that they have detected upon fome occafions a yellow fluid in the veins of the bag, which they have fuppofed to be the yolk in its paffage. Notwithftanding this appearance, which we have fometimes alfo feen, it appears to us moft probable that the yolk is abforbed by the fame fyiftem of veflels which take up the chyle from the fmall inteflines. The yolk-bag and inteftinal canal, as already mentioned, are continuous parts, and fimilarly organized. The yolk-bag is, in fact, a procefs of the inteftines, and fhould be con- fidered as a portion of the body of the chick. It is fup- plied by the fame blood-veffels as the inteftines ; why fhould we therefore deny it the laGeal veffels ? Another argument againft the abforbing funétion of the veins of the yolk-bag may be deduced from the fize of thefe veffels at different periods. During the early part of incubation they are large and numerous ; but after the yolk-bag paffes into the abdomen of the chick, where their abforbing funétion chiefly, if not entirely, exifts, if it does at all, they become fmaller and fewer than the arteries, ‘Vhe veflels of the yolk-bag, when they conftitute the vafeular area, are the organs.of fanguification for the chick, After incubation they appear to become the nutritious veffels of the bag itfelf. It may feem not eafy to affign a ufe for the ductus vitello inteftinalis, unlefs we grant it to be for tranfmitting the fubftance of the yolk into the alimentary canal. If, how- ever, this fhould not be the cafe, ftill a communication would probably exift between the yolk-bag and the in- teftine ; for it is without example, we believe, for the con- tinuity to be interrupted by two adjoining cavities that have a fimilar organization. The dudius vitello inteltinalis we think, theretore, would exift as a‘matter of courfe, independently of any offices it might perform. We have deferibed the purpofes this part ferves in the full grown bird ; it is not, therefore, without its ules, You. XIX. Inc For the hiftory of fortal life ia oviparous aninals in gé- neral, fee Ovum. 7. INCUBUS, in Medicine, from the Latin incubare, to he or fit upon, the fame with the Englith night-mare, fignifies a {tate of imperfect fleep, accompanied with an oppreffion of the breathing, and frightful dreams. By the Greeks this affection was defignated by a name of fimilar import, 22iArnz, ephialtes, from the verb garry L leap upon ; and it has received various other appellations expreflive of the fenfe of weight and oppreffion of the cheft, the laborious refpiration, &c. as well as indicative of the hypothefes by which its occurrence was explained- Thus it has been called 2aiSorry ruypedey epilepfia nofurna, afthma nofurnum, &c.; and by Pliny, /udibria fauni, from he fame notion which dictated the more modera appeliatione of hag-riding, wizard-preffing, mare-riding ; dier mahr, or das reiten der mabr, of the Germans; cochemar of the French; night-mare, &c. This diforder feizes perfons, while fleeping, who imagine that they feel an extraordinary compreffion or weight about the breaft and flomach, which they cannot by any effort fhake off. In this agony they figh, groan, and utter in- diftiné& founds; fometimes they cry out, but more fre- quently they attempt to fpeak, or to move in vain. Thefe feelings give rife to various frightful fuggettions of the imagination: the patient fancies himfelf to be ftruggling with {trong men or devils, or to be in a houfe on fire, or in danger of being drowned in the fea or fome river; and in attempting to run away from danger, or climb up a hill, he fancies he falls back as much after every ftep as he had advanced before. At length the fenfations of oppreffion become intolerable, and the patient awakes ; but the terroe excited by the frightful ideas attending the night-mare, leaves often a palpitation of the heart, with great anxiety and languor, and fometimes a tingling of the ears, and a tremor over the whole body. ; It is altogether unneceflary to attempt an enumeration of the numerous hypothetical explanations which have beer attempted to be given of the ‘phenomenon of incubus, and which have beeidetailed by Awen, Bond, and other writers. (See Awen, Pofit. inaugur. de Incubo. Argentorat.. 1676. Bond, Effay on the Incubus or Night-mare, 1753.) The diforder has commonly been fuppofed to proceed from a ftagnation of the blood in the finufes of the brain, or in the veffels of the lungs, or from too great a quantity of blood being fent to the head. The horizontal pofture, in time of fleep, and the preffure of the Lomach upon the aortz, ina {fupine fituation, have been thought fufficient to occafion a more than ufual diftention of the /imu/es and other -veffels of the brain; and the weight of the heart, prefling on the left auricle and large trunks of the pulmonary veins, may, it is fuppofed, prevent the eafy return of the blood from the lungs, and thus produce an oppreffion and fenfe of weight and futfo- cation in the breaft. (See Bond, chap. ii.) But without entering into a particular examination of thefe opinions, which are far from being fatisfaCtory, we may obferve, with Dr. Whytt, that, if they were true, fome degree of the night-mare ought to happen to every body that hes upon his back, efpecially after eating a full meal. Further, if a horizontal fituation could overcharge the brain with blood, fo as to occafion the incubus, how comes it that people, who remain for fome time in an inverted potlure, do not feel this difeafe beginning to attack them? And why does a flighter degree of the night-mare fometimes feize people who fleep in an erect fituation in a chair? a circumitance which fometimes occurs, not only after eating, but when the flomach is eut of if or AT, Inc erder, and troubled with wind. As the weight of the ftomach, even when filled with food, can have fearcely any effe& upon the motion of the blood in the aorta, fo the preffure of the heart is by much too {mall to be able fenfibly to retard the motion of that fluid in the pulmonary veins, otherwife, people exhaufled by tedious difeafes, who gene- rally lie on their back, would be conftantly afle&ted with the tacubus. We know that certain medicines or poifong worms, and even corrupted bile, or other humours, by difagreeably af- fecting the nerves of the ftomach, produce an oppreflion about the breait, wild imaginations, frightful dreams, raving, and infenfibility ; and there is no doubt that low {pirits, melancholy, and difturbed fleep, often proceed from a dif- ordered {late of the ftomach. It feems, therefore, more pro- bable that the feat of the night-mare is principally in that organ. It is well afcertained that fome forms of epilepfy, and of hytterical fits, originate from diforder in that vifcus 5 and Galen confidered the incudus as a no&urnal or flighter epilepfy.. People troubled with nervous and hypochondriac affections, and who have delicate or flatulent itomachs, are more peculiarly fubject to this diforder ; and it is obferved, that a heavy or flatulent fupper greatly aggravates the night-mare, in thofe who are predifpofed to it. The fym- pathy of the ftomach with the head, heart, lungs, and dia- phragm, is fo remarkable, that there can be no diiliculty in referring the feveral fymptoms of the incubus to a dil- agreeayle ircitation of the nerves of the ttomach. The ineudus is moft apt to feize perfons when lying on their back, becaule, in this pofition, on account of the Ito- mach and other abdominal yifvera prefling more upon the diaphragm, we canvot infpire with the fame eafe as when we fit up or lie on one fide. Further, in that fituation of the body the food feems to lie heavier on the ftomach, and wind in it does not feparate fo readily by the efophagus and pylorus, as in an erect polture, when thefe orifices are higher than the other parts of the ftomach. ‘The night-mare only eccurs in the time of fleep, becaufe the ftrange ideas excited in the mind, in confequence of the difordered feelings of the ftomach, are not then correGted by the external fenfes as they are when we are awake ; nor do we, by an increafed refpiration or other motions of the body, endeavour to fhake off any beginning uneafy fenfation about the ftomach or Breaft. The incubus generally occurs in the firft fleep, and feldom towards morning, becaufe at the earlier period the ftomach is more loaded with food, and that in a more erude and indigefted ftate than in the morning. A leffer degree, amounting only to frightful dreams, is almoft a conftant concomitant of overloaded ftomach in fome habits. See Dream. In fact, if the night-mare were owing to a ftagnation of the blood in the lungs from the weight of the heart, or in the finufes and other veflels of the brain from the horizontal poiture of the bedy, it would become greater the longer it continued, and would f{carcely ever go off {pontaneoufly, But we know that this difeafe, after affecting people for fome time, often gracually ceafes, and is fucceeded by re- trefhinz fleep : for as foon as the load of meat,,or wind, or ether canfe difagreeably affeéting the nerves of the ftomach is removed, the oppreffion and weight on the brealt, wild imaginations, frightful dreams, &c. vanith ; as all thefe pro- ceed originally from the diforder of the ftomach, It may be remarked, however, that, a3 neither flatulency, phlegm, nor crudities in the ftomach, ever produce the fymptoms of hypochondriafis, unlefs the nerves of that organ be indifpofed ; fo neither a horizontal pofture, fleep, nor heavy fuppers, ever IND produce the night-miare, at leafl in any cohfiderdble dee gree, unlefs the perfon be already predifpofed to the com= plaint, from the particular condition of the nerves of the {tomach. : But although the ftomach is the part commonly affected primarily in the cafe of incubus ; yet fymptoms like thofe of the. night-mare may fometimes arife without any fault of the ftomach, when the lungs, or even the brain, are affected. Thus Dr. Whytt obferves, that afthmatic patients, whofe lungs are much obftruéted, are fometimes dilturbed, in time of fleep, with diltrefling dreams, and opprefled with a fenfe of, fuffocation. Startings and oppreffions about the pracordia, with painful dreams, are indeed common occurrences from hydrothorax, chronic coughs, and other pulmonary obftruc- tions; but they are not ftricily analogous to the common night-mare.. Dr. Lower mentionsa patient, who, though he could fleep pretty eafily with his head inclined forward; yet, in the oppofite fituation, he was always foon awaked with horrid dreams and tremors; the caufe of which appeared, after his death, to have been a great quantity of water in the ventricles of the brain. At ail events, a plethoric ftate of habit, by rendering the circulation through the lungs lefe free, may help to produce, or at leaft increafe, the oppref- fion of the breaft in the night-mare. The Cure.—As incubus, then, is only a fymptom of dif- ordered or loaded {tomach, and arifes out of the irritation and, morbid feelings which are thus produced during fleep, the relief of the difeafe, generally {peaking, lies within a nar- row compafs. Temperance in eating and drinking, efpecially at late hours; taking, in fact, either extremely light fuppers. or nofe at all ; and when the dinner is fo late as to be only a fupper with another name, being cautious that that alfo fhould be moderate in quantity, and eafily digettible and un- ftimulating in its nature; drinking thin, fub-acid liquors, where thefe are agreeable to the conttitution, —thefe are the principal remedies required. Brifk ative exercife, by which the digeftive powers may be aided and the ftomach ftrength-. ened, is alfo advifable. It were ufeful, too, to fleep with high pillows, and to lie on the fide as. much as poffible, in preference to the back. If the funétions of the ftomach are much difordered with flatulency, heart-burn, acidity, or opprefiion, with pain, or naufea, after taking food, the re- medies recommended for indige/fion (which fee) muft be refort- edto. The bowels fhould be kept open. See Whytt on Ners vous Diforders, chap. vi. § 18. : INCUMBENT, in Law, a clerk refident on his benefice, with cure. He is called incumbent of fuch church, becaufe he - . ought to bend his whole fludy to difcharge his. cure.. Ses Reoror and Vicar. INCUMBRAVIT.. See Quang incumbravit. INCURABLILE, the name of one of the celebrated. con fervatories or mutic-fcheols at Venice, where orphan gizls, or girls of worthy parents in indigence, ufed to be well educated and regularly bred to mutfic as a profeflion. Ga- luppi was matter of this confervatorio. in 1770, when: the compofition and performance were exquifite. INCURVATION, the aé& of bending a bone, or other body, from its natural fhape. Ixncurvation of the Rays of Light. REFRACTION. INCUS, in Asaiomy, one cf the fmall bones contained in the cavity of the tympanum... See Ear. INCUSSU, in Geography, a town of Africa, in the king- dom of Congo; 80 miles S. of St. Salvador. | ; INDAL, atown of Sweden, in the province of. aes | acre See Licnr, and IND - , *padeap river of the fame name. N. lat. 62° 34’. E. long. IO 23. a INDEA, a town of Africa, in the kingdom of Yemina, ona river of the ike name. ne INDEBITATUS Assumpsit, in Law. See As- SUMPSIT. _ INDECIMABLE, Inpeermasitis, is applied to things not tithable, or which by law ought not to pay tythe. _ _ INDEFEASIBLE, or Inpereazanxe, fignifies what cannot be defeated, or made void. As, a good and indefea- fible eftate. No one, who confiders our laws, conftitution, _and hiftory, without prejudice, and with any degree of at- tention, will affert, that the doétrine of hereditary right implies an indefeafible right to the Englifh throne. See Right of Crown. - INDEFINITE, Ixveterminare, that which has no certain bounds, or to which the human mind cannot affix any. Des Cartes ufes the word, in his philofophy, inftead of in- finite, both in numbers and quantities, to fignify an incon- ceivable number, or number fo great, that an unit cannot be added to it ; and a quantity fo great, as not to be capable of any addition. _ Thus, he fays, the ftars, vifible and invifible, are in num- ber indefinite ; and not, as the ancients held, infinite ; and that quantity may be divided into an indefinite number of parts, not an infinite number. _ Iwnerrnite is alfo ufed, in the Schools, to fignify a thing that has but one extreme: for inftance, a line drawn from any point, and extended infinitely. _ Thus, what they call eternity a partg ante, or eternity a parte pot, ave indefinite durations. ‘ INDEFINITE, in Grammar, is underftood of nouns, pro- nouns, verbs, participles, articles, &c. which are left im an uncertain indeterminate fenfe, and not fixed to any particular time, thing, or other circumftance. _ Inverinite Propofition, in Logic. See Proposition. _INDELAVOY, Enneravoy, or Jndelvai, in Geogra- phy, a town of Hindoottan, in Golconda, lying in the route between the Godavery and Hydrabad ; 75 miles N. of Hydra- bad. N. lat. 18° 26’. E. long. 78° go’. INDELIBLE, formed from delere, to 5/ot, with the pre- pofition in, taken negatively, that which cannot be blotted out, or effaced. Thus baptifm, and the order of the priefthood, are faid to convey indelible characters. INDEMNITY, formed from the negative in, and dam- - num, lofs, in Law, an act by which one promifes to guarantee, or fave harmlefs fome other perfon from any lofs or da- mage that milf accrue to him on any particular ac- count. When a church is appropriated to an abbey, or college, the archdeacon lofeth his indu€tion money for ever; in re- compence whereof he fhall have yearly, out of the church fo appropriate, one or two fhillings, more or lefs, for a pention, as was agreed at the time of the impropriation ; and this penfion is called an indemnity. ' Ixpemnity, with regard to eftates. Inpemnity. See 4é of Grace. INDEN, in Geography, a town of Switzerland, in the Valais ; 18 miles E. of Sion, INDEN-HOTUN, a town of Chinefe Tartary, the ca- pital of the Mantchew Tartars, where they began to efta- blifh their empire over China. N. lat. 41° 46’. E, long. 124° 36. INDENTATIONS of the Coaft f the Ocean, in Geology, or'the peninfulas, headlands, points, bays, gulfs, eltuaries, See WARRANTY. IND &e. which occur on the fhores of moft iflands and conti- nents, are phenomena worthy the attentive confideration of geological obfervers. M. de Luc,in his Geclogical Travels, vol. i. p. 333 and 336, has confidered many of the circum- {tances attending the headlands and gulfs of the northern coaft of Europe, and very properly reprefents, that they arife from the fyftem of vallies and hills, which are feen inland, continuing down much below the prefent level of fea, even to the greateft fathomable depths in fome inftances : and Mr. Farey, who has remarked on this part of M. de Lue’s work in the Philofophical Magazine (vol.xxxvi. p.7.), obferves, that the hills all evidently ran out in headlands into the fea, juft as they would into a fea at a level confi- derably above the prefent, of which a perfe& idea may be obtained by tracing an extended /evel line in any country ; and that the operations of the tides and waves have a power- ful tendency, in mott inftances, to leffen the indentations of the coafls, by wafhing away the projeGing points of hills, and throwing up the alluvial matters therefrom in the inter- vening gulfs, where marfhes are accumulated ; and that the coafts, where cliffs or precipices occur at the headlands only, have not originated, in mott inftances, from faults or depreffions of the {trata under the fea beach, but are occa- fioned by the falling and wafhing away of the points of the hills. It is evident, that where rocky cliffs fringe the bor- ders of the gulfs, as well as the headlands of a diftria, fuch cliffs have a different origin from thofe which are confined to the headlands. See EncroacHMeEnT of the Sea. INDENTED, InpenrTex, in Heraldry, i¢ when the out- line of a bordure, ordinary, &c. is notched, in form of the teeth of a faw. IxpentEp Leaf. See Serrated Lear. INDENTED Line, in Fortification. See Repens. InpENTED Medals and Wheel. See Mepats and Wueen. INDENTURE, in Law, a writing which comprifes fome contraé between two, at leaft ; being indented at top anfwerable to another part, which has the fame contents. See Deep. INDEPENDENT Company andTroop. See Company and Troop. INDEPENDENTS, in Ecckfiaftical Hiflory, a fe& of Proteftants in England and Holland: fo called, as denying not only any fubordination among their clergy, but alfo all dependency on any other aflembly. They maintain, that every feparate church, or particular congregation, has in itfelf radically and effentially every thing neceflary forits own government ; that it has all eccle- fiaftical power and jurifdiction ; and is not at all fubje& to other churches, or their deputies, nor to their affemblies, or fynods. Robinfon, the founder of the fect, makes exprefs ufe of this term in explaining his do¢trine relating to ecclefiattical government : “ Cxtum quemlibet purticularem,”’ (fays he, in his Apologia, cap. 5. p. 22.) ‘efle totam, integram et perfectam ecclefiam ex fuis partibus conftantem, immediate et independenter (quoad alias ecclefias) fub ipfo Chrifle.’* It may probably have been from this very paflage that the title of independents was originally derived. The difciples of Robinfon, originally called Brownijls (which fee), be- caufe John Robinfon, the founder of this fect, was patlor of a congregation of Brownitts that had fettled at Leyden, did not reject the appellation of Independents. It was certaiuly utterly unknown in England before the year 1640: at leatt it is not once mentioned in the eeclefiatlical canons and F 2 and con- ftitutions, iND ftitutions that were drawn up during that year, in the fynods or vifitations held by the archbifhops of Canterbury, York, and other prelates, in which canons all the various fees that then fubfifted in England are particularly mentioned. See Wilkins’s Concilia Magne Britannie et Hiberniz, vol. iv. cap. 5. p. 548. It is true, that not long after this period, and more parti- cularly from the year 1642, we find this denomination very frequently in the Englifh annals. The Englifh Independ- ents were fo far from being difpleafed with it, that they af- fumed it publicly in a piece, which they publifhed in their own defence at London, in the year 1644, entitled “* Apo- logetical Narration of the Independents.’? But in procefs of time, in order to avoid the odium of fedition and anarchy charged on this fe&, the true and genuine Independents re- nounced this title, and called themfelves “* Congregational Brethren ;” and their religious aflemblies “ congregational churches.’” The firft independent, or congregational church in England, was fet up in the year 1616, by Mr. Jacob, who had adopted the religious fentiments of Robinfou. The Independents, though {prung originally from a congregation of Brownifts, were much more commendable than the latter, both in the moderation of their fentiments, and the order of their difcipline. The Brownilts, as we have already men- tioned under that article, allowed all ranks and orders of men promifcuoufly to teach in public, and to perform the other paftoral fun@ions; whereas the Independents had, and flill have, a certain number of minifters, for the moft part regularly educated, chofen refpe&tively by the congregations where they are fixed; nor is any perfon among them per- mitted to {peak in public, before he has fubmitted to a pro- per examination of his capacity and talents, and been ap- proved of by the congregation to which he minifters. The charge alleged againft them by our hiftorian Rapin, (Hitt. of England, vol. ii. p. 514. fol. ed.) who fays, that they could not fo much as endure ordinary minifters in the church, &c. is, therefore, evidently falfe and groundlefs. He was led into his miltake by confounding the Independ- ents and Brownifts. There are other charges, no lefs un- juttifiable, that have been urged again{t the Independents by this celebrated hiftorian, and others of lefs note. Rapin fays, that, with regard to the ftate, they abhorred monarchy, and approved only a republican government. This might have beea true with regard to feveral perfons among the Independents, in common with thofe of other fects; but it does not appear from any of their public writings that re- publican principles formed the diftinguifhing chara¢teriltic of this feéi. On the contrary, in a public memorial drawn up by them in 1647, they declare, that ‘* they do not difap- prove of any form of civil government, but do freely ac- knowledge that a kingly government, bounded by jult and wholefome laws, is both allowed by God, and alfo a good accommodation unto men.”? The Independents, however, have been generally diftinguifhed by the denomination of re- gicides, under a notion that they were chargeable with the death of Charles I. Whether this fa&t be admitted or de- nied, and this is not a place proper for the inveftigation of it, no conclufion can be fairly drawn from the greater pre- valence of republican principles, or from violent proceedings at that period, that can affect the diftinguifhing tenets and conduét of the Independents in general; and efpecially of the feét that bears this denomination in our times. It is certain that cur Independents are fteady friends to a limited monarchy. Rapin is farther miftaken, when he reprefents the religious principles of the Independents as contrary to thofe of all the xelt of the world. It appears from two 4 IND confeffions of faith, one compofed by Robinfon, on behalf of the Englifh Independents in Holland, and publifhed at Leyden in 1619, entitled “* Apologia pro Exulibus Anglis, qui Brownifte vulgo appellantur,” and another drawn up in London in the year 1658, by the principal member of this community in England, entitled «* A Declaration of the Faith and Order owned and praétifed by the Congregational Churches in England, agreed upon and confented unto b their Elders and Meffengers, in their Meeting at the Savoy, O&. 12, 1658 ;”’ as well as from other writings of the In- dependents, that they differed from the reft of the reformed in no fingle point of any confequence, except that of eccle- fiaftical government ; and their religious doétrines were al- moft entirely the fame with thofe that are adopted by the church of Geneva. During the adminiftration of Crom~ well the Independents acquired very confiderable reputation and influence; and he made ufe of them as a check to the ambition of the Prefbyterians, who aimed at a very high de- gree of ecclefiaftical power, and who had fucceeded, foon after the elevation of Cromwell, in obtaining a parliamentary eftablifhment of their own church government. But after the reftoration of Charles II. their caufe declined; and in the year 1691, under the reign of king William, they en- tered into an affociation with the Prefbyterians refiding in and about London, under certain heads of agreement, com- prized in nine articles, that tended to the maintenance of their refpeétive inflitutions. Thefe may be found in the fe- cond volume of Whilton’s Memoirs of his Life and Writ- ings ; and the fubftance of them in Mofheim, ubi infra. At this time the Independents and Prefbyterians, called from this affociation the United Brethren, were agreed with regard to doétrines, being generally Calvinifts, and differed only with refpe& to ecclefiaftical difcipline. But, at pre- fent, though the Englifh Independents and ‘Prefbyterians form two diltin&t parties of Proteftant diflenters, they are diftinguifhed by very trifling differences with regard to church government ; and the denominations are more arbi-~ trarily ufed to comprehend thefe who differ in theological opinions. ‘Che Independents are generally more attached to the tenets diftinguifhed by the term orthodoxy or calvinifm, than the Prefbyterians. Independentifm is peculiar to Great Britain, the United States of America, where it was carried firft in 1620, and by fucceffive Puritan emigrants in 1629 and 1633, from England, and the United Provinces. One Morel, in the fixteenth century, endeavoured to introduce it into France; but it was condemned at the fynod of Rochel, where Beza. prefided ; and again at the fynod of Rochel, in 1644. On the fubjeét of this article, fee Mofheim’s Eccl. Hitt. by Macleane, vol. v. p. 398, &c. 8vo. Neal’s Hift. of the: Puritans, vol. ii. p. 107, &c. vol. iii. p. 547, &c. vol. iv. p- 187, &c. Burnet’s Hift. of his Own ‘Times, vol. i. p- 46, &c. INDERGEREE, in Geography, a river on the N. F. coalt of Sumatra, which runs into the fea. S. lat. 0° 33’, E. long. 103” 20". ; INDERGUR, a town of Hindooftan, in Guzerat% 15 miles EF. of Damauw. INDERMAY Pornt, a cape on the N. coaft of Java. S. lat.96° 12’ %E. long. 108° 18% INDERSKATA, atown of Ruffia, on the river Ural 5 2 miles N. of Gurev. INDERSOUL, a town of Hindooftan, in Baglanas 12 miles E. of Bahbelgong. ) INDER-TAUPLITZ, atown of the duchy of Stiria, on the Enns; 14 miles W. of Rottenman. INDERVA) IND LUA INDERVA, a fmall, ifland.in the Perfian, gulf; so, Ieagues W. of Ormus. ; INDEFERMINATE, in Geometry, is unc rftood o a uantity either of time or place, which has no certain or definite bounds. See INDEFINITE. - ¥ . INDETERMINATE Analy/is, is a particular branch of algebra, in which there is always propofed a greater num- ber of unknown quantities than there are equations; whence, from what we have feen under the article Atcrsra, the queftions become unlimited; but in this {pecies of equations, the folutions muft always be given in integers, or rational fraétions ; and this condition frequently fixes a limit to the number of anfwers that an equation admits of, and even fometimes renders the problem impoffible ; though, in the generality of cafes, the number of folutions is indefi- nite. When, amongft the unknown quantities, there are none that exceed the fimple power, the equation is faid to be of the fir/t degree : when the fecond power enters into the equation, it is faid to be of the fecond degree: and when the third power or cube enters, it is of the third degree ; and fo on. ' Eyery indeterminate equation of the firft degree, as baxtby+tcetdt&ke=o is folved by means of the more fimple equation si ap—bgaHtt. And every indeterminate equation of the fecond degree has its folution, if not abfolutely depending, at leaft intimately conneGted with the folution of the equation . pPo—-Nv=it. Before, therefore, entering upon the general folution of in- determinate problems, it will be proper to confider more particularly the two equations above-mentioned. | Prop. I, To’ find the values of x and y, in the equation axn—by=thw Firft, it may be obferved, that this equation is always poflible under either condition of the ambiguous fign +, provided that a and b be prime to each other ; and without this, the equation is always impoffible; becaufe, in that cafe, the firft fide of the equation will be divifible by the common divifor of @ and }; whereas the fecond fide, + 1, has no divifor; and, confequently, no equality can obtain. But when a and 3d are prime to eachother, the folution may always be obtained by. the following rule: Divide the greateft of the two ani numbers a.and 6 by the other, and then always the laft divifor by the laft remainder, as in the ufual method of finding the greateft common meafure of two numbers, and let the fucceflive quotients arifing from this operation be denoted by @, 8, y, 2, ke. See Common CAsuUre, ' With thefe quotients, placed in one horizontal row, make ?. 7 ° quot'a 6 had. tie. , %a frac’ = pS (e6+1)y te [@e+1)y+a]}? . | i ee Ls, (yB+a)0+@ ~ which may be otherwife exprefled in words: thus, having arranged the fueceflive quotients, as above, the firft fragtion will have « for its numerator, and tr for its denominator ; the fecond will have a @ + 4& for its numerator, and @ for its denominator; and all the other numerators will be found by multiplying the numerator of the laft fraétion by the fol- lowing quotient, and adding th the preceding numera- tor, and the denominators are found exactly in the fame A 77 IND anner, as is evident in the foregoing fractions. Sce Ratio. : rh y o ; it . £x. 1.—Find the(values of x andy, in the equation 16a— 419 = 1, Firk 16)41(2 _ 32 . 9) 16(1 2 9 7)9(% ¥ 2)7(3 6 1)2(2 2 quotients 2, 1, 1, 3, 2 ; ge a AS fraGtions {= 7 Pee Here the laft fraétion but one is 48. and therefore this gives the values of w and y; that is, x= 18 and y= 7, whicle renders 16 4 — 41 y = I, as required. £x. 2.—Find the values of x and y, in the equation: 17x _ sy a= (she Firk 15)17(1. TS 2)15(7 14 1)2(2 2 Quotients se Tac 42 Fractions { ‘of + iv K SP 1. Therefore # = 7, and y = 8, which gives Mxe—isy=— kt _ The demonftration of this rule belongs properly to com tinued fractions, which the reader will find very ably treated of, in the Englifh edition of Euler’s Algebra; and in moft of the French writers on that fubje¢t, particularly in the. «¢ Fffai fur la Theorie des Nombres, par Legendre.”” Having thus fhewn the general method of folving the in-- determinate equation ax — by = + 1; it only remains to make a few obfervations relating to it, and to fhew, that one folution being obtained, an infinite number of other. folutions may be deduced from the one known cafe. In the above examples we found ax — by = + 1, and ax — by — 1, as the queitions required; but we are frequently led to the folution ax — by = + 1, when the queltion requires — 1, and the contrary; which feems at firit to deftroy the generality of the rule ; but this difficulty is cafily furmounted from the following confiderations. Let ap— bg = + », the values of pand g being known, to find, from this equation, the value of x and y, in the equation aw — by= — 1, Since ap — 6g = 1, we have only to make x = dm — p, and INDETERMINATE ANALYSIS. » and y= am — g, andit is obvious that this fubftitution will give a (bm—p) —6 (am—q) =—1;3 and here, by means of the ind ‘terminate m an indefinite num- “ber of values of x and y may be determined. If from one known cafe, asap — bg = + 1, the general values of x and y in the fame equation ax —by = + 1 were required ; we fhould only have to make x = mb + p, and y = ma +q, and we fhould have ftill a (mb + ~p) —b (ma + 9g) =ap—bg= +15 where, by means of the indeterminate sm, an indefinite number of values of x andy may be obtained. We will now illuttrate what has been taught by an example, and then proceed to the more general equations; each of which, however, will -be found to depend upon the one we have been confidering. Ex. 3.—Find the general values of x and y, in the equa- ‘tion Ige—I9gy=k 13)19(4 13 6)13(2 12 1)6(6 6 Quotients 17 Psy be Frations { zi 3, 22 1 2 7 ave have therefore p = 3, andg = 23; and this gives 13p—197=1. Therefore the general values of w and y are z= 19m-+p, and y= 13m-+ gq, or x= 19m + 3, and y= 13m + 2. Affuming therefore m= 0,1, 2, 39 4, &c.3 we have the following values of « and y, m=O) .1, 2, 135 49,151 XE: x= 3, 22, 41,- 60, 79, 98 &e. you? 15s 28, 41, 54, 67 &c. which feries may be continued at pleafure. If the propofed equation had been 13 x — Ig y= —T1;, then ‘having found p = 3, and g= 2, as above; we muit have made x= 19m — 33; and y= 13 m— 2; and then, by afluming m as before, we fhould have 1 Oy Vaibit 2s Bo Ss Ae 5 Tee x = — 3, 16, 35, 54 73. 92, &e y=— 2, Il, 24, 37, 50 63 &e. where it may be obferved, that the fucceffive values of x and y, in both cafes, form a feries of arithmeticals, and may therefore be continued with great facility. Prop. If. To find the general values of « and y, in the equation ax—by=+e. In the firft place we muft have either a and 5 prime to each other, or if they have a common meafure, ¢ mift have the fame, for otherwife the equation will be impoffi! le; and in this latter cafe, the whole equation may be divided by that common meafure, and thus reduced to one in which a and 5 4 are prime to each other: it will, therefore, only be neceflary to confider the quantities @ and 4 as prime to each other. Alfo, after what has been taught in the foregoing propofi- tion, we may always fuppofe that we know the cafe ap—bg = +1; it will therefore be fiifficient in. this place to fhew how the general values of x andy, in the equa- tion ax — by = +, may be deduced from the known cafe ap—bq=xt. In the firft place it is obvious, that fince ap—bqg=+1, we fhall have acp—beq=4te; but this furnifhes only one folution, and in order to have the general values of «and y, we muft fubftitute « = mb + cp; and y ma +¢g; which give a(mb+cp)—b(mat+ecqg) = +3 the ambiguous fign +, in the two values of # and y, being + when ap — 6q has the fame fign with c, but — when it has a contrary one. . Ex. t.—Find the values of x and y, in the equation 9o* — 137 = 10. Firft, in the equation 9p — 13g = +1, we have p= 3, and q = 2, which gives 96 — 13g = +1; and this being the fame fign with 10 in the propofed equation, the general values of x and y are x= 13m+4 3e,0r 33 Jy=om+2 ¢, or x = 13m + 30; y= gm+ 20 And by affuming here m= — 2, — 1, 0, I, 2, &c. we have the following values of x andy: n= 25 — I, Oy, T, 2; 3 ; &e. = Ay 175. 30 43, 56, 69 &e. Y aiaRy IIy 20) .29, 38), 47) (hace each of which values has the required conditions, for 9/24 Slag Sypaieyre % 9. 17 + 1p VEE ae sf 9. 30 — 13- 20= Io 9-43 — 13-29 = 10 ~&c. &e. &e. Ex. 2.—Find the values of x and y, in the equation 7*%— I2y= 19g. Firft in the equation 7p — 129 = — 1, wehavep= 5, and g = 3; where — 1 has a different fign from 19 in i. propofed equation ; therefore the general values of x and y are ea 12m—§-193 4 Sy=7"M—3- 19, 0r Pe PEM = +0 Os J pe oa ae where, by taking m = 9, 10, 11, &c. in order that x and y may be pofitive, we have = 13> 25% 375 49> 61, 73> 85 &c. y= 6°33; 20, 2h. Sqrleat, 45. Gee and in a fimilar manner may any poflible equationax — by = +. be refolved, Prop. III. To find the general values of a and y, in the equation az+by=c. In the foregoing propofition, where the difference of two quantities was the fubjett of confideration, we found that the number of folutions wassinfinite, provided that a and 5 were prime to each other; but in confidering the fum of two quantities, INDETERMINATE ANALYSIS." ¢ quantities, asin thé prefent cafe, the number of folutions is always limited, and in many cafes the equaticn is impoflible ; it may however be demonttrated, that the equation always admits of at leaft one folution, if c > ab —a —4, aandd being prime to each other; and it is propofed in the prefent propotition to afcertain the exaét number of poffib'e folu- tions: that any equation of this kind admits of in integer numbers, and to point out more accurately the limits of poffi- bility. The folution of the equation av + by == c, depends, like that in the foregoing propofition, upon the equation ap — &g = + 1, though its come€tion with it is not fo readily per- ceived. ‘ Let ap —bqg=1; then we have alfo : ac p + bh. dg = ty : and it is evident that we fhall have the fame refult if we make a=cp—mb; andy=cy7—ma; for this alfo gives a (cp —mb) —6b (cg —ma) =e: affuming, therefore, form fuch a value, that eg — ma may become negative, whileeg — m3 remains politive, we fhall have ha a, “Sa (cp— mb) +b (ma—cq)=c3 and confequently « = cp — mb; and y = ma — cq; but if m cannot be fo taken that cq — ma may be uegative, while cf — mbremains pofitive ; it is a proof that the pro- pofed equation is impoflible in integer numbers. _ And on the contrary, the equation will always admit of as many integral folutions as there may be different values given to m, fuch that the above conditions may obtain. Ae we are enabled to determine @ priori the num- ber of folutions that any propofed equation of the above form admits of; for fince we muft have cp > mb, and cg < ma, the number of folutions will always be exprefled by the ore , . c greateft integer contained in the expreffion CP = f) ; ° a as is evident, becaufe m muft be lefs than the firft of thofe fractions, and greater than the fecond, and therefore, the dif- ference between the integral part of thefe fra€tions wil exprefs the number of different values of m3; except when oD, ; Bg - i is acomplete integer, or, which is the fame, we muit con- ey . at chee a fider = asa fration, and reject it, but not-, the reafon a b for which is obvious. Ex, 1.—Required the values of » and y, in the equation: 9x + 13, y = 2000, and the number of poflible folutions in integers. Firk, in the equation 9p — 1349 == 38, we have at once p= 3, and g = 2; therefore the nuniber of folutions: will be exprefled by 2000 X 3 2000 X 2 Si oie re e Te - pera teh yo S87 And thele are readily obtained from the formule x= cp —' mb, or and 4 = ™4 Tye or # = 6000 — 13m; y= 9m — 4000 in which, afluming m = 445, 446, &c.; inorder that 9 m > 4yooo, we fhall have a Hosing félutions, each of which is deduced from. the preceding. one,, by, adding, fuc- céffively 9 to the values of y, and fubtraéting 13. from thofe. of » 5 thus ( @= 215, 202, 189, 176, 163, 150, 137 &c. Y= 5» 4, 235 32) 41, 50, 59> &e.- that is ; 9-215 + 13- S= 2000° Q #202! 4+) PZ ¥4;= 2000 g - 189°4 13 - 23 = 2000. q- 176 + 13. 32 = 2000 &e, &e. &e. Ex. 2.—Given the equation 11x + 13 y = 199, to find: _ the number of folutions, and the values of x andy. Firtt in the equation 11 — 13 ¢g = 1, we have p = 6, and » g= 5; therefore Spo 8 199 Snipes ee ge 13 Li a whence it follows that the equation admits of only one in-- tegral folution, and this is obtained from the formule w= cp — mb,or be hy se 6g oF a= I9a.6 — Wo Ms; y= Ibm — 190.6 where, by taking m= 87, in order that ma — cg may be politive, we have x = 9, and y = 7; which gives IL. + 19-7 = 190, as required. Prop. IV. To find the values of », y, and z, and the number of in- tegral folutions of any equation of the form axt+bytecz=d. Tn the firft place we may obferve, that if any one or more of the co-efficients a, b, or c, be negative, the number of - anfwers is indefinite.. For. let 6 be negative, then the equa tion may be put under the form ax+cz=by+d; in which, by means of the indeterminate y, an infinite num- ber of values may be given to the fecond fide of the equa- tion; and confequently alfo to ».and y. We need, there- - fore, only confider equations of the form above given, in. which the quantities are all conne@ed with the fign +, Now in this equation, as in thofe of the two foregoing: propofitions, if a, 4, and c, have each a common divifor, which d has not, it becomes impoffible ; but if only two of - them, as a and 4, have a common meafure, the equation iss {till poflible, as we fhall fee in what follows. The folution of the equation,. av+tby+ex=d, is refolved by means of the equation ap— bg= + I;> as we have feen is the cafe in the preceding examples. For let one of the three terms, as cz, be tranfpofed to the other fide of the equation ; ther we have ax.+ by = d— ez, in which the values of » and y, determined as in the lat, propofition, will be w= (d—cz)p— mb;-and y= ma— (d—cx) 9, that is, by only fubftituting d —c2 wnhead of ¢, which i¢ the only refpe&. in which this equation differs -from that in the foregoing problem : ‘and here the» only: limits to be obferved are, th, cae < d3 2d, mb < (d—cs)p; 3d,ma >(d—cx)g¢ by-attending.to which, all the poflible values of'x and y ma y INDETERMINATE ANALYSIS, niay be obtained. But+as thefe’ queftions generally admit of a great number of folutions, the objeé of enquiry 1s not fo much to find the values of the intermediate quan- tities, as to determine @ priori the number of them that the equation admits of ; and this, therefore, fhall form the fub- ject of our future inveltigation. Now we have feen, that in the equation ar+by=e, the number of folutions is generally expreffed by cp 8g a pand q being firft determined by the equation ap— b q= 1. If, therefore, in the equation axtby=d—cz, we make fucceffively z = 1, 2) 35 4, &c, the number of fo- lutions for each value of x will be as below ; viz. dp @=94 : ax +4y=d— c, number of folutions a (d—2c)p (d—20)9 Ae 0 Y bs By oe Bes i ES — d—3 d— 3¢ ae tby = d= Bt siois se se = 3p (E= 304 &e. &c. &c. &c. ¢he fum of which will be the total number that the given equation admits of ; and therefore, in order to find the exa& number of folutions in any equation of this kind, we muft firft afcertain the fum of all the integral parts of the arithmetical feries ‘(Gop (d=20)p-, (d—30)p' , (= 4e)P , go a Cees) ea Cae 12g 2 and b é Leg (d— ol are (2=32)¢ 4 (4 49)g eeeend a a a a the difference of the two will be the exa& number of intregal folutions. Now in both thefe feries, we know the firft and laf term, and the number of terms ; for the general term being Sd e8 YP agg = eN9 we fhall have the extreme terms by taking the extreme li- , ; d 7 mits of z, thatisz= 1,.and 2 <—3 which laft value of < c alfo expreffes the number of terms in the feries. Hence then, having the elements of the progreffions given, the fum of the whole feries is readily obtained ; and if therefore we alfo find the fum of the fraétional parts in each, we fhall have, by dedudting it from the whole fums, that of the integral part of the feries as required, The latter part of this problem is readily effefted, for the denominator in each term being conftant, the fractions will neceffarily recur in periods; and the number in each can never exceed the denominator: it will therefore only be neceflary to find the fum of the fraétions in one period, which being multiplied by the number of periods, will give the fum of the fraétional part of. the terms; and thefe taken from the total fum, will give the fum of the.in- tegral part of the feries; amd then, from. what has been before obferved, the difference of the two fums will be the number of integral folutious required. It may alfo be ob- ferved, that when the number of terms do not confift of aw exact number of periods of circulation, the remaining terms or fractions muft be fummed by themfelves, which is. alfo readily effected, as they will be the fame as the leading terms of the firft period ; and it muft alfo be remembered, that ; is to be confidered as a fraction in the firft feries ; a. ; Witte ’ but not ~ in the fecond, as is explained in the foregoing pra« a pofition. Ex. 1.—Given 5 + 7y + 11% = 224, to find the number of folutions that the equation admits of in pofitive integers. ay : 224 Here the greateft limit of z is x rr the equation = = 1, we havep= 3, andg=—2 alfoa = 5, se 4 cm Hi and chien oe Feo ree of which the fums are required, beginning with the leaft term in each, are = 20: allo in” [Jo4 Se ee ee S14 be, » vegeta. 7 7 7 } 24 2.15 2.9 2 ame 2.1 na fe —_— SSS &c. e SS Lee fae Sita ees the common difference in the firft being 3 he ; and in 2.1 the fecond ; alfo the number of terms in each 20. Whence we have 930 for the fum of the firft ; and - ~- 868 for the fum of the fecond. Again, the firft period of fra€tions, in the firft feries, is Swe 4 yO 4 = ch mck jb eee —- =4;3 Ro, eee and in the fecond feries, the firft period of fraétions is 3 2) : = + 0 + Seth tee 5 5 5 5 being confidered as a fra€tion in the firft, but not =25 | — 5 2 in the fecond. Now the number of terms in each feries being 20, wé have 2 periods and 6 terms in the firit feries = 2 x 4 + the firft 6 fra&tions = 11 for the fum of all the fratione ; and therefore 930 — 11 = 919, which is the exa&t fum of the integral terms. And in the fecond, we have 4 pe- riods = 4.2 = 8, and therefore 868 — 8 = 860, which is the fum of the integral terms in this; and hence accord. ing to the rule 919 — 860 = 59; which is the number‘ of integral folutions. Remark.—Simpfon, in his algebra, makes the number of folutions to this queftion 60; but he has evidently intro- duced one (viz. x = 10, y= 14, and % = 14,) which does not obtain. Ex. 2.—Having given 74+ gy + 23% = 9909, it is required to find the number of its folutions in pofitive integers. a 88 the equation 7#— 9 = 1, wehave p= 4, and g = 3; a= 35 . Here the greateft limit of z < alfo in INDETERMINATE ANALYSIS. a= 9, andd=%; allo 9999. — 23. 434 = up» therefore the feries whofe fums are required, will be igo Sem qatelscn(i4xSy 4.9976. —— + &. 1... — Fey pohring a9 ad 2240p 3440 43293 4 ge. ovyi, Be 9908 Bob A 7 7 7 ‘the common difference in the firft being eG 3 > ; aellitel the Secetithe tae =i 9 : alfo the number of terms in each 434 ; that being the greateft limit of x. Hence we have the fum of the firft feries = 963769 and of the fecond - - - - 929349 Alfo the firft period of fraétions in the firft feries, is POM) 90%) BO) gt a GB oot pees ad tebe ee a ee 9 9 9 9 9 9 9 9 5 a= 48 periods,and 2 terms = 5.48 += zi 3 —— 2H 1 9 is 9 Andin the fecond feries, the firft period of fraétions is 2 I pace me. ae eet OT a ee 35 ee te gu! IBD 969 periods; and therefore 62 . 3 = 186. and alfo ——— 3 3 Hence 963769~ — 241 = 3799 HO and 929349 — 186 = 963528 integral terms = 929163 integral terms whence the difference 34365 is the number of in- tegral folutions required. In the foregoing examples we have had two of the terms prime to each other; but when this is not the cafe the fol- lowing transformation will be neceflary. £x. 3.—Let there be propofed the equation, _. I2e@ 4+ 15 y + 20% = 100001, to find the number of folutions. Here no two of the co-efficients are prime to each other, and we mutt therefore proceed as follows. Divide the whole equation by 3, and tranfpofe x, and we have’ bein § z—ii!I 4x +5 y = 33334 —-—7% + which laft muft be an integer ; make Hietetork + Wy and we have z = 3u +13 fubttituting now this value of x, the original equation becomes. 12x + 15 ¥ + 20(3u-+1) = 100001 ; or, dividing by 3, 4% 5 Y p20 U = 33327, the number of folutions in which will be the fame as in the equation propofed, which will be found.as in the foregoing examples, except that here the leaft value of u = 0, becaufe we fhall then fill have 2 = 1, and by proceeding as in the preceding examples, it will be found that the number. ef integral folutions that may be given. to this queition, amounts to no lefs than 1388611. Vou. XIX. Prov. V. Having given any number of equations, fefs than the number of unknown quantities that enter therein, to deter- mine thofe quantities. Let there be propofed the two equations Sige ertea att 8% F ax + hy +cd2x2=d to find the: values of «, y, and a. Multiply the firft by a/, and the fecond by a, whence, by fubtraction, we obtain (’b—adb)y4+ (ac—ac)z=ad—ad' Or, dividing each of thefe known co-efficients by their greateft common divifor, if they have any, and reprefenting the refults by 8”, c", and d", this equation becomes BW y + che = d". Find now the values of y and x in this equation ; and thefe being fubftituted for them in the equation d—czx—by a a will give the correfponding values of x, of which thofe, of courfe, muft be rejeéted that render x fraGtional, and alfo thofe that give (cz + by) >d. £x.—Given the equations 3x%+ S¥+ Te= 560 Ox + 25 "+ 49% = 2920 to find all the integral values of », y, and x. Multiplying the firft by 3, we have Qx+ 157 + 21% = 1680 Qe + 25y + 49 % = 2920 whence 10 y + 28 % = 1240, or Sy +14% = 620 And here the values of y and z are found to be y = 110, 96, 82, 68, 54, 40, 26, 12 % == 5, 10, 15, 20, 25, 30) 355 40 and of thefe, the only two that give Ce eee SS an integer, are as follows, {® 15,7 82, x 15 % 30, y 4O, * 50 which are the only two folutions the equations admit of in integers. The method above given will never fail of producing all the poffible folutions in equations of the above form ; but there are other methods that may be followed in particular cafes which fometimes fhorten the operations. Thefe the reader will find explained in vol. ii, of Euler’s Elements of Algebra. a — — = — — — = — = Prop. VI. To decompofe a given numeral fraction, having a ¢om- polite denominator, into a number of fimple fraQiens having prime denominators, This is, in fa&, only an application of the foregoing prepofitions to this particular problem ; for fet ™ be the P) given fraGion ; and fuppofe, in the firft inftance, that its G deno» INDETERMINATE ANALYSIS. denominator confilts of two prime faétors, or n = a 8, it will then be to find at Poy: 8 eid a + 2 %agt+bp=m in which equation, having determined the values of and q, we fhall have £ + 4 for the fra€tions required; and as thany different ways may any fuch fraétion be decompofed into two others, as the equation a g + 6 p = m admits » then of integral folutions. If the given fraction be ; abe we may firft refolve it into two fractions, and one of thefe into two others ; thus, let Maer et iB, ae en a i =’ then we have abg+cecp=m; and having, from this equation, found the values of p and g, mn axe “Ba 4 ab + c mis > oras+ré= p> find r and we fhall have abe. aa ep, st: Again, let ri , s in this equation, fo fhall we have Minh 5 q abc a i b 13 c” as required ; and in the fame manner may any fra¢tion that admits of decompofition be refolved into others of which the fum fhall be equal to the original fra¢tion. £x.—Whiat are thofe two fractions whofe fum is equal 19-2 Bo —_—_— = Since 35 = 5 X 7, we may make : + ct which produces this equation, 52 4.7 9 = 29: in which the value of = 1, and g = 2 are readily found ; and therefore the fractions fought are = + a pe po i 5 35 N Prop. VII. To find the leaft number, that, being divided by given numbers, fhall leave given remainders. Let .N reprefent the required number, fuch that, being divided by a, a’, a &c. the remainders fhall be refpettively b; 6, 6, &ee-that is, Ne=am+b=a'n+W=a'p + 8" &e. and it is required to find the leaft valne of N, that anfwers thefe conditions. Firft fince . we have am+b=a'n+bB' am — a\n°= }' — 6. Find, therefore, in this equation, the leaft values of m and n by Prop. II.; then will a m + 4, or a! n + Bb, exprefs the leaft number that fulfils the firft two conditions. Let now this number be calledc ; then it is evident that every num- ber of the form aa! g + ¢ will alfo fulfil thefe conditions ; and we muft proceed to find aa'g +c= a"p + Dd! or, adgq—alp=h_—<¢; that is, the leaft value of and g in this equation; fo fhall we have a a! g + ¢ tor the leaft number that anfwers the firft three conditions, and fo on for as many others as may be propofed. £x.—Find the leaft number, that, being divided by 28, 19, and 15, fhall leave for remainders refpetively 19, 15, and II. Here we have 28m + 19 = 19n + 15 = 15 p + 11. Now, in the equation 28 m — 19 n = — 4, the lealt values — of mand nare m = 8, and nm = 123; whence 28 m + 19 = 19 + 15 = 243 5 and it now remains to find 28 . 19 g + 243 = 15 p+ 115 or, 532g — 15 p= — 228. ~ = In which equation » = 512, and g = 14, whence 5329 + 243 = 15 p + 11 = 7691, which is the leaft number having the required conditions. Having thus treated of the folution of indeterminate equa- tions of the firft degree, to as great an extent as our limit will admit of, we fhall proceed to thofe, in which one, at lealt, of the unknown quantities enter in the {quare power, which conititute the clafs of indeterminate equations of the fecond degree ; and as we have. feen that every equation of the firft clafs has its folution depending upon that of the equation a p — 6g = + 13 foin thofe that we are about to inveftigate, the {olution is intimately connected with that of the equation p — N g? == + 1; this, therefore, is what ihail firft engage our attention. Prop. VIII. To find the integral values of » and g in the equation y—-N¢>+h : N being any given number whatever, not a complete {quare. In order to obtain the general folution of this equation, which is always poffible (at leaft with the pofitive fign) we mult fhew the method of extracting the fquare root of any number N, not a complete {quare, in continued frac- tions ; but as this operation belongs properly to the latter fubje&t, we fhall only in this place indicate the method, and muft refer the reader for the demonftration to vol. il, of Euler’s Algebra, and to Barlow’s Elementary Invelftiga- tions. See alfo Square Roor. The transformation of the ,/ N, to continued frations, is performed by means of the following formula. i a be the greateft integer contained in ,/ N; then make egies — 24.—_- 6 = m3 e —— /N+m N — m” oa =u un—m =m; = af n n J N+ ni! . N — m!? , pela u! n! — m! =m; ——— = a" n n i] IR /N+m N — m ; = ul | all nt — ml = ml; —— = nll! n ) n CE Cruanl (Coe. | &ce. &e. In thefe formule u, u', u'', &c. are the greateft integers con- tained in the correfponding fra€tions, which quantities are. the quotients, whence the converging fraétions are to be deduced, by the fame rule as is given in Prop. I. for the quotients «, 2, y, 0, &c.; and by continuing the above ope- rations, we fhall be finally led to a quotient equal to 2 a, at which term we mutt ftop, and the correfponding frations, to the laft quotient before this will give the required values: of p and g. Lx. 7 INDETERMINATE ANALYSIS. Ex. re Required the values of p and g in the equation ane pP-—197=-1. i ' By the above rule, | | a alg.r-o=4; Stay i wa Deg snare ee a) 92-4525 2S = s sat “ 58 2 = Sana 2 5 a YT 3235 x! 2.3—3=3; 2-3 _, ig — 2° cars ra wt | 5-i- 3S 2; = 3 5 — 2 ty 2.2.2 == gg a om oot id tt osaw Ke. &e. Having thus arrived at the quotient 8 = 2 a, we have only to compute the fraétions by the propofition above quoted ; thus, quotients es CO as S3¥ Rh 8% Pe ak eS ee eee ee fractions PSM ap “99 which laft fraGtion gives the values of » and g; that is, p= 170, and g = 39; for 170” — 19.397 = — 1, as required. Ex. 2.—Find the values of p and gq in the equation . f-3¢=—1 Firf, ae. ta) 1 12 — 2” ering SR nce Werte x = 2 oor angie 41-3515 3D =3 af tz HT 13 — 2” a +e elr=—I=2 = 3 é ee es oo ee Ce aes rae a JIZ+ 1 pay I StH &e. &c. Aad having thus arrived at the quotient 2 a, or 6, we have fo" quotients Ri Ola KE Tn 2 fractions a eee a ‘Cee SG Bleed that is, p = 18 and g = 5, which give ; 1 —43.5°= ~—1, as required, ‘ Ex. 3.—Find the values of » and g in the equation Pie FAG sets MBIT Serge ; I Firft al uw +4 2 8 Here we have, at the. firlt ftep, arrived at the quotient 2a, whence the firft fraction, +) is the one fought; for 4’ — 17.1 = — 1, as required. Lx. 4.—Find the values of g and g in the equation zy ne, ye Firft, P 49 v7 14+ 0 14 — 3? tb ge1-0=3;—1 3 = 5 wT, FAB dg eh, = 2 te ESE)? 5 “714+ 2 7 ae oat, : Vitti=: 2 qe pat a5 2 a Tahal 14—3° 5 =) | 5 i 2 = 3; per ‘ pM en nt SOE &c, &e. I Whence we have, ; quotients 3s us 2) fractions 3 his eoic cce ha haat sty which gives 15* — 14. 4° = 1, as required. It will be obferved, that in the foregoing examples we have obtained for each only one folutioa; whereas they all admit of an indefinite number of folutions. Moreover, it does not appear, from the firft three examples, how we fhould have found the values of p and g, if they had beer put equal to + 1 inftead of —1. This fhall, therefore, be confidered in the following propofition. — Prop. IX. To find the general values of x and y in the equation a — Ny?= +1, from one known caflep’' —- Ng’? = +1. In the firft place it may be demonftrated, but our limits will not allow of it, that the equation x7 — N y* = 1 is always poffible for every value of N, providing it be not a complete {quare; and the values of x and y are always deducible, both from s? — N g* = 1, and from p> — Ng’= — 13 but if the operation above given does not produce the equa- tion p> — N g* = — 1, the equation 2» — Ny’ =— 1 is always impoffible. The prefent problem, therefore, di- vides itfelf into three cafes ; viz. to find the general values of w and y in the equation «* — N y’, under the following conditions ; 2 th, —Ny*= 1,from one knowncalep?>—-Ng'= 1 a0, 2° — Noy ss od, pPo-N¢g=-! 3d, x? — N y* = —1, p -Ng=-1 Cafe 1.—Refolve p* — N g* = 1, anda* — N y* = 1 inte (p#av N)(p—gv N)=1t (x+y YN) (w@#-y VN) =1 then we have alfo, (p+ q7N)” (p-qvVvN)"= 1; equating thefe, with the factors in x and y, we have (p+qyN)” x+y JN (p-g/N)"=2-y JN (ptgv N+ (pray ND", q/ N) Ld “aed the factors lf whence x = a tiet 2 / NT p) 4 es Bee and soy N which values of « and y will always be integral, and will be the general values fought ; and thele are evidently indefinite in number, becaufe the power m is indelinite. Cafe 2.—The fame method may be followed here, as in the preceding cafe, except the powers muit be even, 2 2 INDETERMINATE ANALYSIS. in order to convert — 1 into + 1, as is obvious from infpection ; and, therefore, the general values of x and y will be ea Pty Nr + lose ¥ Ne 2 (pte vy N)™— (p- ov N) 2/N Cafe 3.—Hlere, again, we have evidently the fame refult as in the former cafes, except that the power m muft now be odd for every odd power of — 1 = — 1; therefore, if ? ‘ zm* 1 2m4.1 lt WE | Bee ey 2m. umt t | ae AP ea ND Doe cgi 2 J/N Let us now propofe an example in each of: thefe three cafes. £x.1.—In the equation p — 14.¢ = 3, we have p= t5 and g = 4, to find a fecond value of p and g, or of wand y, in the equation x° — 147 = I. I= Make C . Cis 4 7) (15) a Coe a Di ahs — C54 pS BANTAM ui bess { ie B® which give 449° — 14.120? ='1; and other values may be found by afluming any other power above the fecond. Lx. 2.—Given p = 4, and g = I, in the equation PP —17¢° = — 1, to find the values of w and y, an the equation « — I7y? = 4. Here we have, again, f (4+ /17)? + (4 = 497) , d 2 ) y= 44 vai G = v7)" L 2a4fiF whence 33*— 17.8? = 13 and other values’may be ‘found by affuming any other even power inftead of the fecond. = 33 — § Ex. 3.—Given p = 4, and gq = 1, in the equation Pf — 179 = 1, to find the values of # and-y, in the. equa- tion **? — 7 = —k. A ffume : si (4+ WIT) + (= LET) 268 2 | y= 4+ SITY HG yp 6 L ae Z/¥7 Davee whence, 268° — 17. 67*= — 1, and othervalues may be obtained, by affuming other odd powers isiftéad of the third. “ Prop X. To afcertain the poffibility orimpoffibility of every equa- tion of the form x? — Ny* = + A>; and to find « and x in the former cafe, A being < ,,N. The rule for this purpofe, is to convert ,/N into a feries of quotients, as in Prop. VIII. ; andif A be found in the denominator of any of the quotients, that is, if A’be found among{t any of the numbers, ‘which in the propoficion above quoted are reprefented by ‘x, 7', nl", ‘&e. ithe equa- tion is poffible, and the converging frattion correfponding to the quotient preceding this will.give’the values'of and»; but if A be not fo found, then 2s the equation impoffible. This theorem cannot be demonftrated in this place, as it belongs to the theory of continued fractions, and the reader is therefore referred for a proof of the rule to Le Gen- dre’s Effai fur la Theorie des Nombres. con. Ex, 1.—Required the values of « and y, im the equation x” — 239" = 2. Here by the rule Prop. VILE.’ vtjto = 4 I.4-O=> 43 Sot = 7* J 28 Ae SW ‘ se roe eA . =i! 7-1 4 = 33 7 nw 29 + 3 > 3 Having thus arrived at the denominator 2, it follows that the equation is poffible, and the values of x and y are found from the fame calculation as at Prop. VIII.; thus. quotients A» Ty. 2 | fraGtions f he 2 , Bf ( whence x = 5, and y= 1, which gives ~’ sn 239° = 2. Ex. 2-—Requived the values of x and y, in the equation x —I9y' = §- Firft sig + 0 Io — 4? N _ aie I-4—O=4 2a 4 a3 Jig +4 19 — 2? EY 24 275 a a 3 3 4 > 3 5 A LQ 2 Having therefore found the denominator 5, the equation is poflible, and we have quotients 4» 25 1 fra&tions { 4; 9, I 2 whence # = g,:and y = 2, which gives 97 + 19.2° = So as required. Ex. 3.—It is required to afcertain the ,poffibility, er impofhibility, of the equations ; ie apy =e pple Be ILK Yo alate Firft 707+ 0 te wT = 4 | hi 4 TOS $ ad =r ‘17 +4 snot ee I 28 —4 = 5 ; = 5 hes tae, ae 17 "42 25 a) = 8 1.8—4 = 4; sai. Gel whence it follows, that fince only 1 enters into the deno- minators of thefe quotients, no one of the propofed equa- tions are poffible. Cor:—By means of this propofition, we may demonftrate generally the impoflibihty of all equations falling under any of the following forms ; % | m ; SESS, ot HEE E+ > Sp x — (a —a in which A > 1,and << @ Cer, INDETERMINATE ANALYSIS. Cor. 2,—It is alfo deducible from fimilar principles, that ‘the following equations are always poflible; N being a prime number, of the form placed oppofite the refpective equations. ww — Ny? = —1 poffible when N of the form 42 + 1 x —Ny=—2 Bist) 3 ar Su ie . 0 eee sais 3 —Ny= Die teen ae IN e © « ee, % Sa2—I1 -And in a fimilar manner, are deduced the three following theorems : ; 1. IfM and N be both of the form 4” + 3, and not Mx? —-Ny= +1 is always poffible in integer numbers: that is, undey one 2. If M and N be both of -the form 42 +413 then one of the equations Me— Ny=+1 will always be refolvible in integers. 4n + 3, and N a prime number of the form 42 + 1, it will be always poffible to fatisfy one of the equations Mx—M’Ny=+1 Mx? —-MNy=+1 To find the general values of x and y in the equation ¥—-NyY= +A Find the values of mand n in the equation m* — N n* = 1, by Prop. VIII. ; then it is evident that the product and it will be found, upon the developement of the following formule, that 4 (p m+ Nqa)*?— N (pn + qm)? we have, therefore, only to make, x=pm+Nqn x=pm—Nqa or and having before fhewn how to find the general values m and n in the equation m’ — Nn*= 1; it is obvious, that of v and y, in the propofed equation, to any extent at pleafure. inthe foregoing one give p> — Ng* = — ‘A, when’ the equa- tion propofed be +-A; that this may be converted to the m—Nr= — 11. Ex. 1.—Given the values of 'p aud g, in the equation 1 vi values of x and y, in the equation 2 — 7 y" = 2. Firft in the expreffion m* — 7 2° = 1, we have m = 8, R=puntNgoe 38+ 7.1.3 ={Jrgy yopunt gm ©&sg age 1.8 S10r17 y = V7; which give 45° — 9 2997? = 2. And afluming thefe againas new values ofp andy, other values x and y may equal to-each other, the equation or other of the figns + or —. x —MNy? = —1, or 3. If M and M’ be two prime numbers of the form } Nv—MM'y=+1 Prop. XI. from one known cafe p’ — Ng’?= + A. (p' Ny) x (w—Na) = 4A; PNG) x @—N2) = 9 ym — Naa) —N(pn—gm)? I= fH 8 gm J Spite wee by means of thefe formule we may derive different values “Cor.—It appears alfo from this propofition, that if the rule latter fign by means of the equation = 25 vizep = 3, andy = 1, to find the general and n = 3; whence by the above formule'is obtained fo that the feeond values of w and y are, ~ = 45, and be ‘found in infinitum : ov the ‘original values .of p and g may 1 be retained, and new values found for mand a, which anfwers the fame purpofe. £x. 2.—Find the general values of x and y, in the equa- tion 2? — 11y? = 5; the known cafe being p* — 11g* = 5, orp = 4, and g= 1. In the equation m* — 11 n?= 1, we have m= 10, anda = 3, therefore by the formule x=pmtNqn=4.10+ M.1-3=70r73 y= pat gm =.4- 34 b-10 =20r22 that is « = 7, and y = 2, are two-new values of x and y, as are alfox = 73, andy = 22; foreach of thefe give 7 — Il ..2 = § Dk UE 2 ce and in the fame manner, other values:‘may be found to any required extent. As our limits will-not allow of a very full and explicit in- veftiyation of the feveral elegant rules that have-been intro- duced into the indeterminate analyfis by Euler, Lagrange, Legendre, and other diftinguifhed mathematicians ; we mutt refer the reader for the inveftigation @f the methods em- ployed in the following propofitions to the authors above- mentioned, and mutt content ourfelves with barely {tating the operations, without-entering into the demonftration of the theorems themfelves. Prop. XI. Every indeterminate equation of the fecond degree ‘falls under the general formula ax +bxytey+dxe+tcy+ f=o, and this may always be transformed to the more fimple form v—Ar= B. The method of performing thistransformation will be feen from the following partial example, and the formule we have given it being univerfally the fame in all cafes. Affume 6d —2ae= 9; d' —4af=4; (by +d) mai lGy" fies +f) =t;b—4ac= A; Ayt+ ez = 23.9 — b= Then it will be found, by the developement of thefe ex- preffions, that Gx t+bxy t+eyt+dxieyt+f=u'—Ar—B=o, u'— Av? = B. And having found the values of ~ and¢ in this laft equa- tion, thofe of x andy are readily derived in the equation propofed, For becomes u— t—d—45 I= —, and a2 = cee J, or, fub{tituting for y in the laft, we have the following values ; PONS Ms salt 3 b°— 4ac “all (¢ —d) (b*—~4ac) —(u—g)b oa 2a(b —4ac) Ex. 1.—Transform the equation 3x + Bay — gy? + 2x — 5 y = 4110 to its fimpleft form. Herea = 3,6=8,0=>~+3,¢d>2, 8245, fe — P10, Whence ot pee, = AS 100 ABR sa aca igen 46 } d—agafe bi= 1324 g —ASb=B= = v3008y and INDETERMINATE ANALYSIS, and thus the reduced equation becomes u?— 1007 = — 130284. In which, having found nu = 346, and# = 50, Me SEIS aa a We ave (fy aa seer 2 _ t—by—d_ jo—8:-3—2 = = 2a 6 therefore x = 4, and y = 3, are the values of and y in the original equation. It will be obferved that we have employed here the mott general form that equations of this kind admit of, and therefore the formule are more complex than they ufually occur in practical cafes, for when any of the co-efficients a, 4, c, &c. become zero, the expreffions are much fimplified, as appears from the following example. Ex. 2—Reduce 7° + 5xy + y? = 67 to its fimpleft form. Here-a= 9, 6=)4, fe -whence, omitting tho C=..15d = one Ghk — 67, quantities that are zero, we have Bb —4ac= A =29; andg=o d—gaf=h = 1876 g— Ah = 29. 1876= B and thus the reduced equation is u? — 29 f = 29 . 1876". And ina fimilar manner may any indeterminate equation of the fecond degree be reduced to the form? — Ar = B. Having therefore fhewn the method of reducing every in- determinate equation to the form u*3— A# = B; it follows that the folution of this fimple form involves with it the fo- lution of every equation of this kind that can be propofed ; we fhall therefore, in the following propofition, attend to the folution of this particular cafe. But it may be proper to ftate, that there are an infinite number of equations of this kind that are impoffible ; and will admit of no folution, either in integers or fra€tions; and therefore before we proceed farther in the inveftigation, it will be ufeful to lay down a rule, whence their poffibility, or impoffibility, may be af- oo ; as we may thus frequently fave much unneceflary dabour, Prop. XIII. To afcertain the poffibility or impoffibility of every in- determinate equation of the fecond degree. Rule.—Reduce the propofed equation to the form uw— Af = B, and find all the remainders arifing from dividing each of the aaperstgs A—1\? {quares 17, 2°, 3°, 4°, &c. (==) by A; and alfo the remainders arifing from dividing each of the f{quares 17, 27, 3”, 4°, &c. —) by B; and again, divide the greateft of thefe numbers A, B, by the lealt of them, and obferve the remainder. Then if B be greater than A, this lalt remainder will be found amongft thofe of the upper feries ; and the number A will be found amongft thofe of the lower feries, if the equa- tion be poffible. And converfely, if thefe conditions have not place, the propoied equation will admit of no folution, either in in- cegers or fraétions, Again, if A > B, then the remainder arifing from ee will be found in the lower feries, and the number B in the upper feries of remainders, if the equation be poffible; and if thefe conditions have not place, the propofed equation is iffl- poflible. : ; ne iy Note.—It is to be obferved, that equations falling under the poflible form are not always folvible in integers, the proof extending only to their folvibility in rational numbers, which may therefore fometimes be fra¢tional ; but when they fall under animpoffible form, they will admit of no folutions either in integers or fraétions. ; Ex.— Required the poffibility or impoffibility of the equa- tion an 1 i Wy? sz TT fquares 1,\, '2%,° 3* dividedomy oy remainders Ty | Wye, e {quares Iy 25 3 43..% Give is remainders Ty Bs, 4! “oes “a i — = 1 and remainder 4. a nS f Now 4 is found in the upper feries of remainders, but 7 is not found in the lower; therefore the equation cannot have place either in integers or fra€tions. And for the fame reafon, the equation 4° — 7 y? = 11 2? is alfo impoffible, for if this was poflible, fo would like- wife i — 7 cs z 7 fible. Ex. 2.—It is required to afcertain the poffibility of the equation 11; which we have feen is impof- poflibility or im- @ — 13 9" = 12> 6° divided by.1g {quares Big 2 9135) Gay 5%, remainders I, 4) Q) 3, 12, 10 And the fame fquares, divided by 12, give for remainders I, 4; 9) 4, I, © alfo, Eo 1 and remainder 1. And here, fince 12 is found in the upper feries, and Lin the lower, the equation is folvible. Note.—If the equation propofed be of the form v—Ay= —B, we mutt employ, inftead of the pofitive remainders arifing from A, the negative remainders of the fame, that is, taking the quotients in excefs. Andif the equation have the form a+ Ay = B, then we muft employ the negative remainders of B. Hav- ing thus given an idea of the method of judying of the pof- fibility of every equation of the form «?— Ay? =B; and having alfo fhewn how any indeterminate of the fecond degree may be reduced to this form, it only remains to fhew the method of folution of the above equation; or, which is {till a more general form, of the equation Ag = Bazi; but in this, asin the other propofitions, we can only indicate the method, without attempting to invefligate the rationale of the operation ; as this would carry: us much beyond our. limits, Now it is fhewn under the article DiopHanTINne, that a gate the INDETERMINATE A'NALYSIS. the folution of the equation «* — y? = Cz? is always to be obtained ; and in the following propofition, with which we fhall conclude this article, it will be feen that every equation of the form x — Ay* = Bz’, which is poflible, may be transformed to another of the form x’? — y’ =¢z"; and that x, y, and 2, in the original equation, will be dependent upon thofe of x, y', and z', in the transformed equation; and therefore thefe laft being known, the former will be known alfo. ny a) PROPAKIV. To transform every poflible equation of the form x — Ay’ = B, to another dependent equation-of the form al? ne y” cg alt, . ' Ex, 1.—It is required to transform the equation 2* — 5 y* = 11 x, to another of the form x”? — y®=c2". Having firft afcertained the poffibility of the equation by the foregoing propofition, the transformation may be effect- ed in the following manner. Affume « = ny — 11)’, and this fubftituted for x, gives If Take n, fo that n*— 5 may be divilible by 11 ; that is, let 2 = 4, and our equation becomes yp —8yy' + iry” = 25 or y-4/P- sy =e or, by making y — 49'= x, we have Pe cyt ee ar, a? — 2° = 5 y", asrequired, that is, the equation has been reduced from the form 2* — Sy? = 11 2’, to another of the form #” — y?=c2", or, at leafl, to x” — z*= 5 y”; which differs from the forego- ing only in the letters. And, by means of the values of 2’, ‘y', and z, in this lait, we readily arrive at thofé of wv, y, and x, inthe one propo- fition ; for «’= y — 4y'; Or y= 2’ + Gy! “and w= ay— ily’, or w= 4y— 119! Now we have feen, under the article DiopHantine, that the general values of x'.and z, in equations of the form wi — 25 y", are 7 f= PT 5 eT aga P- FF y= 2p4 whence we have, for the general values of x, y,:and 2%, in the eqnation propofed, . x= 4p + 209 + 10 pq pire Hott 5 rt 8 pg M9 pict? Ges where ~ and g may be aflumed any numbers-at pleafure. If p= 3,-and g = 1, we have.x = 86,'y = 38, andz = 4; which numbers aifwer the required conditions; for 907 — 5; SO ak. 4"; and by giving different values to p and g,-a variety. of other integral values may be found for ., y, and x. Note.—1t may happen that the firft transformation will not reduce the equation to the form required, in which cafe | we miuft again transform this anew, and, by continuing the operation, the reduétion to the final form will be ultimately ‘effected, Ws Ex. 2.—Required the values of x, y, and, in the equa- tion im 129? = 13 2. _ Firft, we have x =ny — 13y', and {ubftituting for «* we have n—2\ , ; ad, ae ae —2nyy +13 =2 5 and here 2 = 5, whence the equation becomes y — Oy y'’ +139" = x5 OF y— 55) — yh et Make now y — 5 y' = x', and it becomes aft — 2% — 72 y” and the general values of x' and x in this equation, are” w= 3P +49, %=3fP —4 7, andy'= 299. Whence, by afluming p = 2, and g = 1, we havex' = 16, - z = 8, and y'= 4; {fo that in the original equation, the va- lues of x, y, and x, are as follow ; wiz. I) Bae 5, I=. 3° fre 9 = 5" — 129’ = 1327 Of, ee 2 the latter values being formed by dividing the former by their greateft common divifor ; and either of thefe fets of numbers anfwer the required conditions ; for, 132°— 12. 36° = 13 . 8’, and 33° —12.9° = 13. 275 and various other values may be obtained by changing the- values of p and g. We will now give one example in which the required re- duction does not take place in the firft transformation. Ex. 3.—Required the values of x, y, and z,-in the equa- tion x= Fy = 192. Affume, as before, « = ny — 19 y': then the fubftitution of this value for x, gives (= yi —2ayy'+19y"=2°; inwhicha= 10 Whence 5x — toyy’'+19y"= 2, or- 25y'— 50y y' + OF y"= 52 Making now, 5 y—107'= x! we have Ee ee eee and here, though we have ‘not arrived at the form required, the lait co-efficient is reduced from 19 to 5; and thus every fucceffive transformation will reduce the co-efficients, till we ultimately arrive at that of unity ; but without purfuing this reduction farther in the prefent cafe, we are led to the folu- tion in aneafier manner: for we fee immediately, that 2’ may be afflumed = 5,°y'= 2, ands = 12. ; And from thefe we readily afcertain thofe of a, y, and <,- in the propofed equation ; for .. ' i ‘ Bit ny 5 +10.2 Qe SG Ory 2 —_> = 5 4 x=any— Igy; or « = 50 — 38= 12 s.= | >; or s= ‘which give 12° 5.5% = 19. 3’, as required. We fhall now conclude this article with a fynopfis of in- determinate formule ; the demonttration and application of ewhich the reader will find in Barlow’s « Elementary Inveiti- gation.of the Properties of Numbers,’’ Synopjis INDETERMINATE ANALYSIS. Synoplis of Indeterminate Formule. Equation ax -- by = +c | General valueof x= mb+cq of y= matcp in which expreflions m is indeterminate, and the values of pand g refult from the folution of the equation ap — ‘ h CAS ew Form. i. Form. 2. Equation ax + by = ei sale value of x = ¢ qg—mb ) =ma— Cp Number of folutions =f ch a the quantities p and g being afcertained as above, alfo m indeterminate. Form. 3. Equation ax + by tex =d General value of w = (d — ¢z) g,— mb y =ma—(d—cx)p the quantities and g being found as above, alfo m inde- terminatc, and 2 any integer <—. c Form. 4. Equation x* — ay? = 2, Gen. val. of x = p> + aq’ Lar 22 F z= p —ag in which expreffions a is given, and p and qg are indeter- minates, that may be affumed at pleafure. Form. §. Equation x? + ieee a Gen. val. of x = p* — aq* Z= p+ aq a@ being givens and p and g indcterminates as above. Form. 6. Equations ax* +bxy + 9? = 2? ee val. of a = 2pq + bg LL Ag, z=ptbpqt+aq, where p and q are indeterminates, and a and 3 given quan- tities. \ Form. 7. ' Equation aa* + bv = x, Gen. val. of x = Bi A i ns ag ae see, | NR A! & where a and / are known quantities, and p and g indeter- minates, Form. 8. Equation m* x? + bv +¢ = 2% *—¢ Gen. vakiof x = —f_ 4%. bq'—z2 mpg mp + meq —bpq* bg —2z2mpq Z : where m, b, and ¢ are any given numbers, and p and g inde- terminates, —— Synopfis of Indeterminate Formule. Form. 9. Equation ax* 4 ba +m? = 2% Gen. val. of « = bg —2zmpg pa ag. mp+ amg —bpq where m, a, and 6 are known ; and p and g indeterminates, = Form. to. Equation «* —Ny? = + 1. (o+9 VN)" + (p—g aN)” Gen. val. of x = 2 _ (p+ 7 /N)"—G-¢ VN een 2 ie spb where p and g arife from the equation p* — Ng? = + 1, and m is indeterminate ; except that it muft be even or odd, as the cafe requires. See Prop. 1X. Form. 11. Equation x» — Ny* = + A, ics val. of x = pm+Nqn Pore na the values of mand x being formed from the equation m’ — Nn = + A, and and g from the equation p? — N g* SS Sap i. Form. 12. Equationax?+ ba" +ex4f?= 2% 6" 4 Pf" eet pot (8 af-—4 bf etc’) Bf" = (4.0 f° Pai e)? ’ where all the quantities a, }, c, and fare given. Form, 13. Equationaxt+baitea+da2e+f =x. : Sbfi+—4acdf? + d’) 8f? ee 2 be 4 df er } Partie val. of x 160 f= 64 af? Bed f5 di where a, 6, c, &c. are all known quantities. Particular val. of * = Or, Form. 14. Equation m? e'+ba3+cat+dx+e=xx% ; w == 16c* m*+—64 em®—8 cb? m?+ 5 | Partic. val. of Hsia cba bate b) mz? where alfo m, 5, c, &c. are all known quantities. Form. 15. Equation m xt + bx? +c 4 dx Se ag Particular val. of + = Or; x In thefe expreffions m, 6, c, &c. are known quantities, and with regard to the ambiguous fign, it muft be obferved, that when it is taken + in the numerator, it muft be — in the denominator, and thé contrary. | Form. 16. Equation a x3 Eat fe ae f° ae (c2 gb fs digg: 27 af° a Gog 4, b, c, &c. being known quantities, { Particular val. of x = Synopfit IND Synopjis of Indeterminate Formule. Form. 17- Equation m?#? +42? +cxr+d=2'. Ge - gb? = 27 dam { Particular val. of x = Gin _ ou @, b,c, &c. being given quantities, as above. Form. 18. Equation m’x? + bx’ tex + fi =2% ee 3 bf") in. 24m f° — cs 7 b* — 27 f m® Particular val. of « = Ny “= (36m'—b) 9m Wik) a Or, cot menpert where alfo m, b, c, &c. are known quantities. Form. 19. Equation x” + avy + by? = x3. General val. of # = # — btu? — aby’ y= 3 uU+ Zatu + (@'—s)x} %2=t4+ atut+ x’, where ¢ and » may be aflumed at pleafure. Form. 20. Equation x* + by*= x’. Gen. val. of # = ¢? — btu? y= 3tu — bu3 Ries Fy ep ee?, t and u being indeterminates, as above. Form. 21. Equation «* + by? = z+, Gen. val. of « = ¢'— 6b#?u? + 4 u4 y=4tu— 4btw a PAU at # and uw being indeterminates, as above. Form. 22.. Equation z* + dy* = x”. Gen. val. of « = #7” —@t™-? uw’? hb 4 3t"-4u4 B°— &e, { y= at ae — yt b+ et" ub’ &c. wat? + bu, where ¢ and u are indeterminates, and 1, « 6, y, 0, &c. the eo-eificients of (¢ + ~)”. Form. 23. Equation x3 + cy? = %*. Gen. val. of « = 4t*— 4ctu? y = 8H u + cut w= 4+ ¢u', where ¢ and uw are indeterminates. Form. 24. Equation #* + aa°y + bxy* + cy? = 2, ur? + 2auw + (a? —b) w* { Pantie. val f= — 2w Gen. val. of x = #? + 2euw + acw* y= 2tu—2buw— (ab—c)w’, where u and w are indeterminates, on which alfo depends the value of ¢ INDETERMINATE Problem, is that in which there are more unknown quantities than there are equations ; and therefore from the principles of algebra the number of fo- utions is unlimited, unlefs fome other condition enter, fuch Vou, XIX. IND as requiring integral values of the unknown quantities, by which means the problem frequently admits of only a defi- nite number of anfwers; as may be feen in the preceding article. In geometry a problem is always faid to be inde- terminate, or unlimited, when there are not fufficient data to limit its con{ftruétion. INDEX, in Anatomy, the fore-finger. INDEX, in Arithmetic, is the fame with what is otherwif= called the characteriftic, or exponent of a logarithm. The index is that which fhews of how many places the abfo- lute number belonging to the logarithm confitts, and of what nature it 18, whether an integer, or a fraction. Thus, in this logarithm 2.521293, the number ftanding on the left hand of the point is called the index ; and be- caufe it is 2, it fhews you that the abfolute number anfwer- ing to it confifts of three places; for it is always one more than the index, becaufe the index of 1 iso; of ro is 1; and of 100 is 2, &c. And, therefore, in thofe {mall tables of Briggs’s Logarithms, where the index is omitted, it muft be always fupplied before you can work by them. If the abfolute number be a fraction, the index of the logarithm has a negative fign, and is marked thus, 2.562293 : which fhews the correfponding number to be a decimal fraction of three places; wiz. 1.365. Mr. Townly has a peculiar way of noting thefe indices, when they exprefs fractions, now much in ule, viz. by taking, inttead of the true index, its arithmetical complement to 10; fo that he would write the logarithm now mentioned thus : 8.562293. How indices are to be added and fubtrated, fee under the article LoGARITHM. Invex, in Mufic, a character ufually placed at the end-of each line to indicate the firft note of the next line. The following is the chara&ter W, which the Italians term a mofira, the Englifh call it a dire@. Invex of a Book, is that part annexed to a book, referring to the particular matters or paflages therein contained. InpeEx of a Globe, is a little fty'e fitted on to the north. pole, and turning round with it, pointing to certain divi- fions in the hour-circle. It is fometimes alfo called gnomon. See Grose. InnEx of Relation, in Algebra, the fame with fcale of re- lation. See SCALE. Ixpvex, or Indice, is alfo the denomination of a congre- gation at Rome, whofe bufinefs is to examine books, and to put fuch as they think fit to prohibit the reading and {felling of, into anindex. See CONGREGATION. Indices, or Expurgatory Indicés, denote the name by which the catalogues of prohibited books are called ; among which, however, there is this difference, that fome are condemned purely and abfolutely, and others only donee corrigantur, ull they be corrected. Father Paul fays, that Philip of Spain was the firft who, by a law made in 1558, procured an index to be publifhed of the books condemned by the inquifition of Spain. Pope Paul IV. took the hint, and ordered the congregation of the holy office at Rome to print a fecond in 1559. But Gret- fer, (De Jure Prohib. lib. li, cap. 19.) fays, that the index of prohibited books . was printed in Italy, by the papal autho- rity, in 1548; anda larger one in 1552; and another larger in 1554. Pius LV. recommended the matter to the council of 'l'rent; the fathers at Trent, after feveral debates not being able to agree, thought fit to refer the whole aflair to the pope, who, with the advice of certain learned prelates, , publihed IND publifhed an index of the prohibited books, and certain rules, by his letters, in form of a brief. The fame Pius IV. ina bullof March 24, 1564, fays, that the index was pre- {ented to him by. order of the fynod, that it fhould not be publifhed before it was approved by him. It was accord- ingly publifhed by him, with a ftrict charge that it fhould be received by all, and the rules prefixed to it obferved. Clement VIII. enlarged and confirmed it, together with the rules, and commanded it to be publifhedin 1595. This was called by the name of the Roman index. By the fourth rule, the common reading of the Holy Scriptures is forbid in thefe words. Since it is plain by experience, that if the facred writings are permitted every where, and without difference to be read in the vulgar tongue, men, through their rafhnefs, will receive more harm than good ; let the bifhop or inquifitor determine, with the advice of the parifh priett or confeflor, to whom to pérmit the reading of the bible, tranflated by Catholic authors in the vulgar tongue, according as they fhall judge whether it be moft likely that fuch reading of the {eripture may do harm, or tend to the in- creafe of faith and piety. Let them alfo have the fame ower as to all other writings. But if any, without fuch leave, fhall prefume to read or have them, without firlt fhewing the bible to the ordinaries, he fhall not receive the abfolution of his fins. And as to all bookfellers, who fhall fell the bibles tranflated into the vulgar tongue, without fuch leave, or by any other method fhall publifh them, let them forfeit the price of the books, and let the money be given to pious ufes by the bifhop ; and letthem be fubject to other punifhments ; at the pleafure of the faid bifhop, according to the nature of the offence. As to regulars, they fhall not read or buy them, without leave firft obtained from their prelates.” ‘The tenth rule enje‘as feveral reftrictive regula- tions concerning the printing and publication of books. Hardouin’s Concil. tom. x. p. 207, &c. - In pope Clement’s catalogue is an extraordinary kind of decree, that all the books of Catholic authors, written fince the year 1515, which was the year immediately preceding that in which Luther began to declaim again{t indulgences, fhould. be corrected ; not only by retrenching what is not conformable to the doétrine of Rome, but alfo by adding what may, be judged proper by the correctors. After this, the duke of Alva procured another to be printed at Antwerp in 1571, which was publifhed by Francis Junius about the year 1586. There were two others publithedNn 1584, and 1612, by the cardinals Quiroga and Sandoval, and feveral others by the inquifitors. and matters of the facred palace. The moft confiderable of all the in- dices is that of Anthony a Sotomayor, fupreme prefident and inquilitor-general in the kingdom of Spain, which was made for all the ftates fubjeLio the king of Spain, and comprehends all the others. This was publithed, with the advice of the fupreme fenate of the general inquifition, in 1640, and reprinted at Genevain 1667. ‘fo this there were many rules prefixed; and to the Geneva edition was added the index of the decrees which were made by the mafter of the holy palace, by virtue of his office, or by the command of the holy congregation, or by the holy congregations for the indices and holy office, after the before-mentioned index of the council of T'rent.- "Fhe rules of the former indices are explained and confirmed by-thefe ; and the fifth rule, which efilarges the fourth of the index of Trent; prohibits not only all bibles in the vulgar tongue, comprehending all except thofe that are Hebrew, Greek, Latin, Chaldee, Syriac, Ethiopic, Perfic, and Arabic ; but all parts of them, either printed or manufeript, with all fummaries and abridgments IND in the vulgar language or tongue. Limborch’s Hift. of the Inquifition by Chandler, book ii. chap. 16. — - : Of the operation of thefe indices, the authors of the En- cyclopedia obferve, that there has been hardly any. good book of piety or morality in their language which has not been profcribed. * Art. Judex. . a Ixpex of the variation and of the inequability of curvature. See VARIATION of curvature. INDGEH, in Geography, a town of Afiatic Turkey, in Caramania ; 12 miles S.W. of Kaifarieh. Inpcen Su, a town of Afiatic Turkey, in Natolia; 40 miles S.W. of Angura. a4 INDIA, an extenfive country of Afia, which was di- vided by Ptolemy and the ancient geographers into * India extra Gangem”’ and India intra Gangem,”’ orthe peninfula beyond the Ganges and that on this fide of the Ganges, to which the appellation of India, according to its original etymology, as the country of the people called “¢ Hindoos,’® properly belongs. (See Hinpoosran.) India on this fide of the Ganges was bounded, according to Ptolemy, on the W. by the Paropamifus, Arachofia, and Gedrofia; on the N. by mount Imaus ; on the I. by the Ganges ; and on the S. and partly on the W. by the Indian fea. The gelf into which the Indus difcharged itfelf was called «Canthi-Col- pus,”’ and an ifland in this gulf was named « Barace.”’ The wef{tern mouth of the Indus was denominated * Sagapa.’” Ptolemy enumerates feven mouths of this river under this and the following names; v2. ‘ Oftum Sinthum, Oreum, Cha- riphi, Saparages, Sabalaffa, and Lomibare.” ‘To the Ganges Ptolemy affigns five mouths, viz. thofe of * Polura, OfGum Magnum, O. Chambericum, O. Pfeudo-ftomum, and Anti- bole.””? The part of India, which was fituated towards the weit, was called ‘* Indo-Scythia.”’ India beyond the Ganges had the Ganges on the W., on the N. certain parts of Scythia and Serica; on the E. the country of the Sine ; and to the S. the Indian fea. ‘This, however, is a vague name for wide and various, and till of late very much unknown, regions between Hindooftan and China. The territory thus denominated is rich and extenlive, and includes the Birman empire, and the dominions of Pegu, Siam, Laos, Cambodia, Siampa, Cochinchina, ‘Tonquin, and Malacca: which fee re{pectively. See alfo Arracan, Asam, Awa, and Trizer. For an account of “ Hither India,’’ or “ India intra Ganges,’’ fee Hinpoosran and Laft India COMPANY. For an elaborate account of the “ Political Hittory of India, from the introduction of Mr. Pitt’s bill, A. D. 17845 to the prefent date,” fee Mr. Malcolm’s “ Sketch.” Lond. 1811, ovo. _Ispra, La, Company, Coins, Silk, &c. See COMPANY, Cor, SIpK, ae: INDIA, SIGISMONDO D’, in-Biography, a mufical compofer,, who was born at Palermo in Sicily, and flourifhed abont the year 1610. In 1611 he publifhed at Venice two fets of madrigals; and 3n 1627 a book of motetti, which did not go down the ftream of oblivion unnoticed. : INDIAN, ia a general fenfe, denotes. any thing belong- inig to the Indies, Eaft or Wett. INDIAN Arrow root, 1 Botany. See MARANTA. Inpian Lay, in Geography, « bay that lies on the W. fide ” of Bonavilta bay, in Newtoundland, Ixpian Lread, in Botany. See CASSADA. INDIAN Corn. See Maizz. Inpian Corn-ears, Petrified, in Natural Hiffory. Many of the eavly writers on extraneous foffils mention ears of Indian corn among their reliquia.. Mr. James Parkinfon, in his Organic Remains, vol. i. p. 450, fhews, that what Emanuel Swedenborg ‘ IND Swedenborg figured as an ear of Indian corn, is the ftem of an unknown plant, which, at pages 424 and 427, is faid to belong to the feries of vegetable remains inclofed in iron- ftone nodules. See plate 14. fig. 4. of W. Martin’s Pet. Derb. Inptan Cove, in Geography, a harbour on the E. coaft of Indian ifland, in Dufky bay. Inptan Creek, a creek on the ifland of Antigua, a little to the welt of Standfalt point.—Alfo, a river of Virginia, which runs into the Ohio, N. lat. 40° 25’. W. long. 80° 40! Alfo, a river of Virginia, which runs into the Chefapeak, N. lat. 37° 43'. W. long. 76° 27. Inpian Crefs, in Bojany. See Tropmoium. Inpian Corn Falls, in Geography, a cataract in the river St. Mary, about twenty miles from lake Superior. N. lat. 46° 22'. W. long. 84° 25'. Ixpian Fig, in Botany. See Cactus. Inpran Gem, in Mineralogy. See Gem. Inpian God-tree, in Botany. See Ficus, Inpian Head, in Geography, a cape on the E. coatt of ~ New Holland, fo called by captain Cook from the number of natives who appeared near it as he pafled in May 1770. S. lat. 25° 3'. E. long. 123° 56!. Inp1awn //land, a {mall ifland near the coaft of North Ca- rolina, at the mouth of Pamlico found. N. lat. 35° 23'. W. long. 76° 50’. -- Alfo, an ifland onthe S. part of Dufky bay, on the coaft of New Zealand, about four miles in circum- ference, fo called by captain Cook; about four miles from Pickerfgill harbour. Inpran Ink. See Ink. Inp1an Leaf, Malabathrum, in Botany. See Tamara- PATRA. rn Ixpran Mallow. See Siva. Invian Millet. See Hotcus. Inptan Oak. See Trcrona. IwpiAn Ocean, in Geography. See Ocean. _ Iypran Old Town, a town of America, in Hancock county, ftate of Maine, fituated on an ifland in Penob- fcot: river, jult above the Great Falls, and about fixty miles below the orks. Here are about 100 families of Roman Catholics, the remains of the Penobfcot tribe, and the only Indians who refide in the diftriét of Maine. They are in a thriving ftate, having obtained from the Provincial congrefs a tract of land, 12 miles wide, interfeéted in the middle by the river, and vefted with a right of hunting and fifhing as far as the mouth of Penobfcot bay extends. They have a decent church, and a prieft who conduéts their reli- gious fervice. Inpraw Reed and Shot, in Botany. See CANNA. Inpran River, in Geography, a river of Delaware, which runs into Rehobot bay, N. lat. 38° 40’. W. long. 75° 16’. —Alfo, a fmall arm of the fea, between Chandlin and Plea- fant river, in the diltri€t of Maine.—Alfo, a river on the E. coaft of the peninfula of Eaft Florida, rifing near the fea- coaft, and running from north to fouth, and forming a kind of inland paffage along the coaft. It is alfo called Rio Ays. N. lat. 27 30’. W. long. 80° 40°.—Alfo, a town in Suflex county, Delaware, containing 1547 inhabitants. InpIAN Sound, a gulf or bay on the S. coalt of Patagonia, communicating with the Straits of Magellan by St. Jerome’s channel. Indian Town, afmall town of North Carolina, 52 miles from Edenton. Ixpian Town Point, a cape on the eaft coatt of the ifland of Antigua. N. lat. 17° 15'. E. long. 61° 22’. Isp1an Wood, called alfo Jamaica and Campeachy Wood, is taken out of the heart of a large tree growing plentifully IND in the ifles of Jamaica, Campeachy, &c. ufed in dyeing, its decoétion being very red. It has been obferved, that putting fome of this decoétion into two bottles, and mixing a little powder of alum with the one, it will become of a-very beautiful red, which will hold ; the other in a day’s time becoming yellow, though beth bottles were {topped from the air alike ; and that if a little of the fame decoétion be expofed to the air, it will become as black as ink in the fame {pace of time. See CAMPEACHY Wood, Brasit, and Locwoop. INDIANA, in Geography, a territory on the N.W. fide of the Ohio, and lately formed a part of the N. W. territory. In January 1801, it was by aét of congrefs erected into a temporary government, with powers and privileges fimilar to other territorial governments. It lies between the Great Miami river E., the Miffiffippi W., the Ohio S., and the II- linois N. It is divided into the three following counties : Counties. {nhabitants. Chief Towns. Inhabitants. TE ROK,.